EDIT: I should have read the other posted answer before writing the answer below. Obviously, my suggestion
that the name "induction" was coined by Poincare is wrong. I am curious, then, as to when the name "induction" gained popular currency. Was Poincare simply taking an established term and turning it to his own purposes in the philosophy of mathematics?

In one of his essays (I forget which one, and don't have the reference at hand) Poincare
discusses mathematical induction in the formal way that we think of it, and explains that
it is this principle that allows mathematical argument to escape the rigid confines of formal tautologies and take flight on mathematical intuition. In fact, he uses the name induction in deliberate analogy with inductive reasoning in science (to be contrasted with the deductive reasoning that underlies logical manipulations of definitions).

I don't know how much of his contribution to the formalization of induction is original,
and how much he is building on earlier work. The wikipedia article on induction has a small
amount of history and mentions Boole, Peano, and Dedekind (all working in the 19th century) and does not mention Poincare, while the wikipedia article mentions Grassmann as well.
This suggests that Poincare is indeed building on their
earlier formulations. (Aside: I didn't see in either article a statement as to where the precise statement induction originated (in a footnote quoting from Boole in the induction article, the term
induction does not seem to be used), so it seems conceivable that the actual name "mathematical induction" comes from Poincare.)

The wikipedia article on induction mentions Bernoulli as an earlier employer of the "inductive hypothesis".
It also mentions the well-known "infinite descent" arguments of Fermat,
which are a variation on induction (in fact, they are a direct appeal to the well-ordering of the natural numbers), and mentions several earlier examples, going back to ancient times. None of these earlier examples are explicitly applying induction in our modern sense, though; rather, they are making arguments or calculation which are implicitly of an
inductive nature.

Summary: I hope that someone who knows more history and has more sources than wikipedia at hand will give a more definitive answer, but my guess is that, while inductive style arguments date back to the beginning of mathematics, the precise logical formulation of inductive arguments dates back to the 19th century (and represents part of the concern for logical foundations that developed in that century), and that the actual name "induction" may originate with
Poincare (in the early 20th century).