As a bit of a placeholder response before an actual expert comes along with a wall of text, I can give you a rough outline of trends. First of all, there did exist something like the concept of a "court philosopher" in the Hellenistic period (so from the death of Alexander the Great in 323 B.C. to the end of the Roman Republic). These were figures more or less professionally trained in philosophy who were hired by wealthy families to, well, hang around and philosophize.

This must be understood in the context of Hellenistic philosophy overall, when philosophy took a turn (so the traditional narrative goes) from the more "pure" and speculative philosophy of classical Athens to the elaboration of more practical, self-oriented techniques of living. This is when Stoicism and Epicureanism emerged as popular movements, because the focus of their philosophy was on providing practical guidance to living a better life, decreasing anxiety, increasing pleasure, etc. Naturally a wealthy person would be more inclined to pay someone who had concrete guidelines for living than someone who was continually engaged in metaphysical speculation.

Prior to this, there were the Sophists, whom you can witness first-hand in the Platonic dialogues. They charged students money to teach them philosophy and rhetoric among other things. (Socrates likely also charged some of his students money, despite Plato's picture of him. See Aristophanes' The Clouds, where Socrates seems like a bit of an unsavory charlatan.)

Sources: A.A. Long. Hellenistic Philosophy. University of California Press, 1986.

Pierre Hadot. Exercices spirituels et philosophie antique. Paris, Etudes augustiniennes, 1981.