I realize this question borders on not qualifying as answerable or mathematical enough, but I would suspect it relevant somehow. I'll remove it if it's not. If you look at some explanations of mathematical induction you can find authors first choose to point out that mathematical induction isn't inductive, in the sense of inductive reasoning, which indicates the term as initially confusing. Would it work out better to rename "mathematical induction" as "mathematical recursion" or have I missed some subtly of how the term "recursion" usually gets used?
A distinction is often made between (mathematical) induction and recursion, according to which the former is a proof technique, while the latter is a method of defining and constructing mathematical objects.
But even ignoring that distinction, which not everyone makes, the name mathematical induction is far too well established to be worth trying to change. First, it does no real harm: students should learn early on that the meaning of ordinary words used as technical terms can’t reliably be predicted from the everyday senses of the words. (Normal, anyone?) And any attempt to change it would almost certainly be futile anyway.
I believe the name comes from the fact that P(n) "induces" the truth of P(n+1) for proposition P.
Dictionary definition of induce (one of several): To bring about, produce, or cause: That medicine will induce sleep.
Let me know if I'm wrong, but when I make my inductive hypothesis I believe I'm doing induction, making a generalization based on an observation ("assume the base the base case holds for a particular n=k...)
Why would you say it's not inductive? What author are you referring to?
And if you agree with what I said than the induction part is more important than the recursion part, because without the induction, you couldn't perform the recursion.
Edit: Your comment does make sense. In that case, this might answer your question: '"Mathematical induction" is unfortunately named, for it is unambiguously a form of deduction. However, it has certain similarities to induction which very likely inspired its name. It is like induction in that it generalizes to a whole class from a smaller sample. In fact, the sample is usually a sample of one, and the class is usually infinite. Mathematical induction is deductive, however, because the sample plus a rule about the unexamined cases actually gives us information about every member of the class. Hence the conclusion of a mathematical induction does not contain more information than was latent in the premises. Mathematical inductions therefore conclude with deductive certainty.'
In other words, you can't check every value, but you assume it must be true for every value, and that's a generalization. Hope that helps.