This is actually answered pretty straightforwardly in The Silmarillion, in the essay "Of the Rings of Power and the Third Age":
And Curunír (that was Saruman the White) was chosen to be their chief, for he had most studied the devices of Sauron of old. Galadriel indeed had wished that Mithrandir should be the Lead of the Council, and Saruman begrudged them that, for his pride and desire of mastery was grown great; but Mithrandir refused the office, since he would have no ties and no allegiance, save to those who sent him, and he would abide in no place nor be subject to any summons.
Saruman was well known for his learning (as opposed to wisdom): he knew a lot, and he had studied Sauron's works. The essay "The Istari" in Unfinished Tales1 adds that "he was regarded by well-nigh all, even by the Eldar, as the head of the Order," echoing Gandalf's statement in The Lord of the Rings (Book II Chapter 2, "The Council of Elrond") that "Saruman the White is the greatest of my order."
Gandalf, on the other hand, was a wanderer at heart; he didn't want to be in charge of anything, but to go where (apparently) his heart led him. From the phrasing of the "Rings of Power" passage, it appears that he actually was offered the position, but turned it down.
1 This essay was written just after The Lord of the Rings and was intended to go in its index, if Professor Tolkien had been able to inflate the indexes to the size he had wished for.