Is there a generic name for 'performing an operation on both sides' of an equation/inequality?
Yes. The name is algebra (maybe "doing algebra" is more appropriate and/or sounds better).
Here there are three sources:
- The word "algebra" comes from a book written in 830 by the astronomer Mohammed ibn Musa al-Khowarizmi (c. 825), titled Al-jabr w'al muqâbala. The word al-jabr meant "restoring," in this context, restoring the balance in an equation by placing on one side of an equation a term that has been removed from the other; thus if $-7$ is removed from $x^2 - 7 = 3$, the balance is restored by writing $x^2 = 7 + 3$. Al' muqâbala meant "simplification," as by combining $3x$ and $4x$ into $7x$ or by subtracting equal terms from both sides of an equation. [...] When
al-Khowarizmi's book was first translated into Latin in the twelfth century,
the title was rendered as Ludus algebrae et almucgrabalaeque, though other titles were also used. The name of the subject was finally shortened to algebra. (Kline's book)
- He [al-Khwarizmi] did this in his book al-jabr w al-muqabalah. “Al-jabr” (from which stems our word “algebra”) denotes the moving of a negative term of an equation to the other side so as to make it positive, and “al-muqabalah” refers to cancelling equal (positive) terms on both sides of an equation. These are, of course, basic procedures for solving polynomial equations. Al-Khwarizmi (from whose name the term “algorithm” is derived) applied them to the solution of quadratic equations. (Kleiner's book)
- The mathematical sense [of the word algebra] comes from the title of a 9th-century Arabic book ilm al-jabr wa'l-mukabala, ‘the science of restoring what is missing and equating like with like’, written by the mathematician al-Kwarizmi. (Oxford Dictionaries)
@BillDubuque pointed out that a correct answer for the question should give a name for the following rule: equalities are preserved by "performing an operation on both sides" (see the comments in this post).
In the Patrick Suppes terminology, the name of this rule can be Consequence of the Rule Governing Identities.
Rule Governing Identities (RGI): If $S$ is an open formula, from $S$ and $t_1=t_2$, or from $S$ and $t_2=t_1$ we may derive $T$, provided that $T$ results from $S$ by replacing one or more occurences of $t_1$ in $S$ by $t_2$. Moreover, the identity $t=t$ is derivable from the empty set of premises. (Suppes book)
Remark 1: Given an operation $f$, let $S$ be the formula $f(z)=f(z)$. Then, by the RGI,
In words, "equalities are preserved by performing an operation on both sides".
Remark 2: The RGI also justifies general substitutions in equalities. For example, $x+y=2$ and $x=y+3$ implies $(y+3)+y=2$ (here, $S$ is the formula $x+y=2$). To understand the word "general" see the comments in the ASKASK's answer.
Remark 3: Other names (probably, the usual ones) are Replacement, Substitution Property and Substitution by equality (see other answers and comments).