Why the name "square root"? Why do we say that $\sqrt{a}$ is a square root of $a$?
Is this because $\sqrt{a}$ is a root of the function $f(x)=x^2-a$?
Cubic root similarly?
Thanks in advance
 A: This question appeared recently at the English Language and Usage StackExchange (here). I will reproduce here the answer I gave there.
In short, root is the translation of the Latin word radix, which is itself a mistranslation of the Arabic word jadhr. That word has multiple meanings in Arabic, one of which is indeed root. But the Arabic-speaking mathematicians1 who introduced it used it in its other meaning of 'basis', 'foundation', 'lowest part'.
1I say 'Arabic-speaking' because some of them were non-Arabs writing in Arabic, the lingua franca of their time and place. For example, Muḥammad ibn Mūsā al-Khwārizmī, who is traditionally regarded as the founder of algebra, was Persian. In particular, the word algebra comes from the Arabic word al-jabr, 'balancing', which appears in the title of his best-known work.
Discussion
According to Encyclopaedia Britannica,

In the 9th century, Arab writers usually called one of the equal factors of a number jadhr (“root”), and their medieval European translators used the Latin word radix (from which derives the adjective radical).

But why did the writers of mathematical treatises in Arabic use the word jadhr?
The reason has to do with the distinction between 'concrete' and 'abstract' numbers (see e.g. here), which was apparently very important to the mathematicians of the time:

concrete number: a number referring to a particular object or objects, as in three dogs, ten men
abstract number: a number that does not designate the quantity of any particular kind of thing

                                                                                             Collins Dictionary, here and here
(Nowadays these issues are usually relegated to the sciences and engineering, under the name of dimensional analysis.)
It seems that there was a tradition of thinking, going all the way back to ancient Egypt, that, in order to compute e.g. the area of a rectangle, one cannot simply multiply two abstract numbers denoting the lengths of the sides of the rectangle. Rather, one of these numbers has to first be multiplied by the 'basis' of area (I am not sure if there is any conceptual distinction between this sort of 'basis' and the modern concept of a unit of measurement). It is this 'basis' that Arab-speaking mathematicians referred to as jadhr, the Arabic word which means 'basis', 'foundation', 'lowest part'. However, jadhr also means 'root', and many translators of the Arabic words into other languages (including Latin and Chinese) mistakenly thought that jadhr was being used with that meaning in the texts. Here is the relevant excerpt from a source I will quote more fully below:

Thus the Egyptian computed the square by ﬁrst multiplying the one side (9 Kket) in the square unit. This was the square—basis (jadhr) to be multiplied by the other side, representing the pure number. The same process is also described by al-Khowarizmi who says: “And in every square ﬁgure one of its sides multiplied into the square unit is its jadhr … and we make the other side, namely, hj, three, and this is the number of its jadhr’s.”
                                                 S. Gandz, On the Origin of the Term "Root." Second Article,
                                                        the American Mathematical Monthly, vol. 35, p. 74, 1928.

Extended discussion
The following is taken from the article On the origin of the term "root" by S. Gandz (The American Mathematical Monthly vol. 33, 261–265, 1926; available as open access here):

The term “root” has its origin in the Arabic. “Latin works translated from
the Arabic have radix for a common term, while those inherited from the
Roman civilization have latus.”2 Radix (“root”) is'the Arabic jadhr, while latus (Greek, πλευρά, pleura, meaning “rib” or “side”) is the side of a geometric square.
2See Smith, History of Mathematics, vol. II, p. 150.
It is certainly rather strange that such a term as “root” should be used in this
connection. It suggests that if the basic number is a root, the square might be
a bush, and so on up in a kind of a mathematical garden.
The Chinese, indeed, do use the word kun to mean root, grass, and shrub, and
the Hindus also use the word mūla for the root of a plant, but this was very
likely due to the Arabic influence, which is so often seen in China and which
may have spread into India by way of China. It is, however, a fact that thus
far we have no satisfactory explanation as to why this botanical term should
have found place in the theory of numbers. …
Therefore the problem before us, as already set forth clearly by Professor Smith, is to to find out whether the medieval Latin authors were correct in their translation of the Arabic word jadhr by radix (“root”). …
The writer therefore started out to investigate the real meaning of the word
word jadhr, not depending upon lexicons or the Latin versions of the thirteenth century, but seeking the meaning as it it appears in the manuscripts of the old Arabic mathematicians themselves.
Muḥammad ibn Mūsā al-Khwārizmī (c. 825), is the oldest Arabic mathematician of much prominence, and it was his ’Ilm al-jabr wa’l mnuqabalah that gave to Europe both the name and the foundation principles of algebra.  His chapter Bab al Misacha ("on geometry”) begins as follows: “Know that the meaning of the expression ‘one in one’ is an ‘area’; and its meaning is one cubit (in length) in one cubit (in breadth). And every roof of equal sides and angles which has one cubit for each side is a (square) unit. But if it has two cubits for every side and has equal sides and angles, then the whole roof is four times the roof which has one cubit in one cubit. … And in every square roof of equal sides one of its sides (multiplied) in a square unit is its jadhr; or if the same be multiplied in two (square units) then it is like two of its jadhrs, whether this roof be small or great.” …
Jadhr not only means “root” but also “basis,” “foundation,” “lowest part.” Mohammed ibn Musa … begins his chapter on areas by introducing the new notion of a square unit. Then he says that in order to get the area of any ﬁgure we must ﬁrst multiply the one side by the square unit; this is then the basis to be multiplied by the other side. We do not multiply one side by one side, but one jadhr, representing the square basis, by one side representing the number. The same deﬁnition and idea is also to be found elsewhere in his algebra. …
The term [jadhr] does not mean “root,” but “square basis,” that by whose multiplication we get the square area. This was the reason why jadhr was used by later writers, such as Omar Khayyam, as the basic number of a square number. The latter did not know anything more about the original meaning of jadhr, and he used the word loosely for dil‘ (Greek, πλευρά), which means “rib” or “side.” But he, like al-Khowarizmi, still knew that it was a concrete number in opposition to an abstract number.
By the time of Beha Eddin (c. 1600) the original meaning was entirely forgotten. In his Kholdsat al-Hisdb (“Essence of Arithmetic”) he says: “What is multiplied into itself is called jadhr in arithmetic, dil‘ (“side”) in geometry, and shai’ (“thing,” “cause”) in algebra.” He certainly understood by jadhr the abstract “root.” In this misunderstood form the term jadhr found its way into medieval Latin as radix.

And in On the Origin of the Term "Root." Second Article by the same author (The American Mathematical Monthly, vol. 35, 67–75, 1928; here), we find a further explanation:

It is now interesting to ﬁnd that this rather strange process of squaring the area is preserved in the Rhind Mathematical Papyrus as the oldest method of computing the area used by the Egyptians as early as 1650 B.C. Thus it is quite possible that in formulating this geometric deﬁnition of the jadhr al—Khowarizmi was prompted, not only by mathematical reasons, but also by old historical reminiscences of the Egyptian geometry.
The Egyptian unit of length in the measurement of land was the Khet of 100 cubits. The commonest unit of area was the setat or square Khet which contained 10,000 square cubits. For practical purposes in land measuring they used a unit called the cubit—of-land or the cubit—strip. It was a narrow strip of land 100 cubits long and one cubit broad. Smaller portions of a setat were expressed in such cubit-strips. To get the area of a rectangle they sometimes multiplied its length in cubits by its width in Khet. This gave them the correct answer in cubit strips. Still, more clearly we see in Problem 48 that for the multiplication of 8 Khet into 8 Khet and of 9 Khet into 9 Khet “the working actually written looks like the multiplication of 9 ‘setat by the pure number 9.”
Thus the Egyptian computed the square by ﬁrst multiplying the one side (9 Kket) in the square unit. This was the square—basis (jadhr) to be multiplied by the other side, representing the pure number. The same process is also described by al-Khowarizmi who says: “And in every square ﬁgure one of its sides multiplied into the square unit is its jadhr … and we make the other side, namely, hj, three, and this is the number of its jadhr’s.”
This archaistic way of computing the square ﬁnds its justiﬁcation not only in the “peculiar Egyptian system of multiplication” but in the very nature of primitive computation. The ancient Egyptians did not compute areas and measure their ﬁelds according to abstract rules. They originally found the area in an experimental practical way by taking a small square measure and trying out how many times it was contained in the ﬁeld to be measured, “as we today measure cloth by the yard.” For this purpose the mere length-measure, the Khet or cord, could not be of any use. They had, therefore, to create the square-unit, the cubit-strip, being a Khet long and one cubit broad. This was the ﬁrst jadhr, square—basis, used in practical life.
This cubit-strip might also, in all probability, be the underlying idea for the Egyptian conception of the square root. The idea of the square root existed in Egypt and the technical term for it was Knbt, literally “corner” or “angle.” Peet has the explanation for that, that the length of each of the two sides of a square containing any corner of it was its square root. Yet it is not the pure, one-dimensional side which contains a “corner” but only the jadhr, the side multiplied with the square unit, contains a “corner.” “Corner” or “angle” is the primitive word for a small square unit. Apollonius (c. 225 B.C.) deﬁnes the angle as the contracting of a surface at one point under a broken line. Since the word “corner,” “cornerstone” usually implies also the meaning “lowest part,” “basis, foundation” we might see in the Egyptian Knbt the source and origin of the Greek pythmenes (bases), the Arabic ass and jadhr, the Hebrew iqqar and ash and the Hindu mūla.
Thus we are enabled to pursue the origin of our term “root” to an algebraical
term denoting the concrete number with a basis, the ﬁrst power (x) and to a
geometric term denoting the basic square unit (x·12). All the three conceptions are contained in the Arabic jadhr and are preserved in the deﬁnitions of al-Khowarizmi.

A: I suspect that it's because if you have a square of side $x$, its area is $x^2$. Therefore if you're given the area and asked to find the side, you have a problem to solve: "What thing, under the action of multiplying by itself, produced this area $A$?" Perhaps someone decided that that thing is the "root" from which grew the area.  
Similarly for cubes. 
A: The "root" of "square root" is from latin radix.
From Florian Cajori, A history of mathematical notations (1928), page 361 of I vol of Dover reprint :

The principal symbolisms for the designation of roots, which have been developed since the influx of Arabic learning into Europe in the twelfth century, fall under four groups
  having for their basic symbols, respectively, $R$ (radix), $l$ (latus), the sign $\surd$, and the fractional exponent.
The sign $R$; first appearance.-In a translationa from the Arabic into Latin of a commentary of the tenth book of the Elements of Euclid, the word radix is used for "square root." The sign $R$ came to be used very extensively for "root," but occasionally it stood also for the first power of the unknown quantity, $x$. The word radix was
  used for $x$ in translations from Arabic into Latin by John of Seville and Gerard of Cremona. [...]
With the close of the seventeenth century [the symbol $R$] practically passed away as a radical sign; the symbol $\surd$ gained general ascendancy. 

See page 366 :

Origin of $\surd$.-This symbol originated in Germany. L.Euler guessed that it was a deformed letter $r$, the first letter in radix [see page 213 of II vol : L.Euler says in his Institutiones calculi differentialis (Petrograd, 1755), p.100 : "in place of the letter $r$ which first stood for $radix$, there has now passed into common usage this distorted form of it $\surd$."]
This opinion was held generally until recently. The more careful study of German manuscript algebras and the first printed algebras has convinced Germans that the old explanation is hardly tenable. [...] The oldest of these is in the Dresden Library, in a volume of manuscripts which contains different algebraic treatises in Latin and one in German. [...] They [the main facts found in the four manuscripts] show conclusively that the dot was associated as a symbol with root extraction.
Christoff Rudolff was familiar with the Vienna manuscript which uses the dot with a tail. In his Coss of 1525 he speaks of the Punkt in connection with root symbolism, but uses a
  mark with a very short heavy downward stroke (almost a point), followed by a straight line or stroke, slanting upward. As late as 1551, Scheubel, in his printed Algebra, speaks of points.

See page 144 :

In 1553 Stifel brought out a revised edition of Rudolff's Coss. Interesting is Stifel's comparison of Rudolff's notation of radicals with his own, and his declaration of superiority of his own symbols. We read: "How much more convenient my own signs are than those of Rudolff, no doubt everyone who deals with these algorithms will notice for himself. But I too shall often use the sign $\surd$ [...]."


Added
From John Fauvel & Jeremy Gray (editors), The History of Mathematics: A Reader (1987).
Page 229 :

Al-Khwarizmi (ca.780 – ca.850) : "A square is the whole amount of the root multiplied by itself."

Page 250 :

Luca Pacioli (1445–1517) regarding quadratic equations : "Now we must see in how many ways they can be made equal, one to the other, and the other to the one, and two of them to one of them, and one to two of them. On this I say that they can be made equal to each other in six ways. First, the square to the things. 
  Second, the square to the numbers. Third, thing or things to numbers. [...]"

Page 260 :

Gerolamo Cardano (1501 – 1576), Ars Magna, on the cubic equations : "Cube the third part of the number of 'things', to which you add the square of half the number of the equation, and take the root of the whole, that is, the square root, which you will use, in the one case adding the half of the number which you just multiplied by itself, in the other case [...]."

Finally :

René Descartes (1596 – 1650), Geometry, page 299 of 1637 edition (see Dover reprint) : "if I wish to extract the square root [racine quarrée] of $a^2+b^2$, I write $\sqrt{a^2+b^2}$; if I wish to extract the cube root [racine cubique] of $a^3 - b^3 +abb$, I write $\sqrt{C.a^3 - b^3 +abb}$ [...]"

