It is a translation from the German Führer (which also is the reason that in older literature, as well as a fair bit of current literature, the conductor is denoted as f in various fonts). Originally the term conductor appeared in complex multiplication and class field theory: the conductor of an abelian extension is a certain ideal that controls the situation. Then it drifted off into other areas of number theory to describe parameters that control other situations.
Of course in English we tend not to think of conductor as a leader in the strong sense of Führer, but more in a musical sense, so it seems like a weird translation. But back in the 1930s the English translation was leader rather than conductor, at least once: see the review of Fueter’s book on complex multiplication in the 1931 Bulletin of the Amer. Math. Society, page 655. The reviewer writes in the second paragraph "First there is a careful treatment of those ray class fields whose leaders are multiples of the ideal..." You can find the review yourself at http://www.ams.org/bull/1931-37-09/S0002-9904-1931-05214-9/S0002-9904-1931-05214-9.pdf.
I stumbled onto that reference quite by chance (a couple of years ago). If anyone knows other places in older papers in English where conductors were called leaders, please post them as comments below. Thanks!
Concerning Artin's conductor, he was generalizing to non-abelian Galois extensions the parameter already defined for abelian extensions and called the conductor. So it was natural to use the same name for it in the general case.
Edit: I just did a google search on "leader conductor abelian" and the first hit is this answer. Incredible: it was posted less than 15 minutes ago!