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 
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.  Back in the 
1930s the English translation was leader rather than conductor.  See the review of 
Fueter’s book on complex multiplication in the 1931 Bulletin of the Amer. Math. 
Society, page 655.