Cokernel: Inspiration for it's name? The cokernel, as currently defined, is the set of "linear constraints". (Wording due to Jonathan Zhu.)
What is the reason for the name cokernel?
Intuitively, a cokernel should complement a kernel. This does not seem to be the case.
Indeed, intuitively, you would think the cokernel corresponds to the image of an operator, as the dimension formula and other considerations would suggest. Similarly the co-image should be the kernel.
Is there a more compelling reason for the nomenclature?
Aside:"Its" really should be "it's”. Apostrophes to possessives. Let "its" be it's own word denoting "it is”.
Of course, I'm well-familiar with definitions from linear algebra. I do not know category theory.
Perhaps, a mismatch of attributes.
Edit: I'm not sure why my comments are being deleted. Some of them are included below.
I do not lack mathematical maturity. I am a Harvard Mathematics PhD student on medical leave. I believe to abstract too much is to be stuck in the midgame some days. When does abstraction become obfuscation?
 A: Mummy already gave out the idea but let me add some details (which required  to put this as an answer and not a comment). Instead of thinking of "co" as related to "complement", we should in fact think of it as related to "quotient", at least in the following cases: let $W \subset V$ be a subvector space, or a subgroup or an ideal. One has the following short exact sequence
$$ {0} \longrightarrow W \overset{\iota}{\longrightarrow} V \overset{\pi}{\longrightarrow} V/ W \longrightarrow {0}$$
(image of a map has to be equal to the kernel of the next map, example at "V", $\operatorname{Im}\iota = \operatorname{Ker}\pi$. The condition at "W" and $V/W$ give injectivity of $\iota$ and surjectivity of $\pi$).
Hmm..., in fact I'm not sure at all about the following as I did not use categories for a while now... In some categories, these short exact sequences split (or maybe if they split on the left, then they also split on the right etc.). To make it short, I' trying to say that we have $V = W \oplus V/W$ and any complementary to $W$ will in fact be isomorphic to $V/W$ which is why in the first years, it seems unnecessary to introduce quotient, at least in this example. (That splitting on the left and splitting on the right is the abstract way to say that in this case, complementaries and quotient can be identified)
Reversing arrows give formally similar universal properties but their interpretation are in fact often very different.
Another remark: a map is injective if its kernel is reduced to 0 (neutral element), it is surjective if its cokernel is 0.
Remark to the aside: the "s" does not come from the verb to be, it comes from declination, it is the form of the genetif, cf. german.
