William Shakespeare was responsible for the line 'Jesters do oft prove prophets,' in King Lear. This was said by Regan in Act V scene iii. She has been poisoned by her sister Goneril. The sisters' show cruel and selfish motivations as they fight over Edmund.
There is a tradition in drama of the 'wise fool' and Shakespeare utilised this. It is likely that this concept is where your friend drew his/her idea from. The fool in King Lear is astute and honest with his king - one of the few characters who does demonstrate nobility and loyalty in the play. The character of Feste in 'Twelfth Night' displays similar qualities.
While your friend may have missed the originator of this thought, he is accurate in that Shakespeare expressed the import of the expression. Many a word of truth is spoken in jest is an expression that goes back farther in time to Geoffrey Chaucer who, around 1390, wrote the Canterbury Tales. In the "Cook's Tale." the line is as follows:
Be nat wrooth, my lord, though that I pleye. Ful ofte in game a sooth [truth] I have herd seye!
In Middle English: "But yet, I pray thee, be not wroth for game (don't be angry with my jesting)
A man may say full sooth (truth) in game and play.
Later, William Shakespeare expressed this idea in a manner closer to our contemporary version in King Lear (1608):
Jesters do oft prove prophets. (V,iii,73)