When you enrich someone with money in the hope or expectation of having a favor from him in return, you “grease his palm.”
The present expression has been in vogue since the beginning of the eighteenth century, changed since the early sixteenth century only in the substitution of “palm” for “hand.”
Our present form, however, is a direct translation of a French phrase of the Middle Ages, “oindre la paume a quelqu’un.”
Littre, the French philologist, tells of an ancient story about an old woman whose two cows had been seized by the provost and who then received the advice that she would have saved herself from trouble had she first “greased his palm.”