The history of court jesters is a bit jumbled, not least because their functions varied between different eras and societies, but also because what little source material we have mostly consists of anecdotes and witticisms that a given court's various public and private chroniclers saw fit to write down. Many anecdotes do involve "speaking truth to power," but usually moderated by the jester's ongoing attempt to remain both useful to the monarch on the one hand, and alive and employed on the other. Sometimes this attempt fails spectacularly, as when Archy Archibald, jester to James I of England and by all accounts not very good at his job, told him that he thought the Prince of Wales was more popular with the common people; the ensuing father-son quarrel resulted in Archy being "tossed in a blanket 'like a dog'" by the Prince and his friends. (Archy was later fired by James' successor, Charles I, for accidentally revealing sensitive political information in one of his offhand remarks.)
In general, court jesters provided many benefits to their monarchs. A monarch with a court jester might be more likely to be perceived as close to the common people (due to the people's enjoyment of comedy), wealthy and powerful (due to the luxury of having a jester at all, and the jester's ability to skewer nobles or the church or anyone else while under the monarch's protection), and also magnanimous and forgiving (if the jester was able to poke at least mild fun at the monarch without having to fear execution or banishment). Having a jester also contributed to at least the perception that a ruler was not surrounded by yes-men; Elizabeth I allegedly rebuked one of her fools for being insufficiently severe with her.
But did they ever actually serve as advisors, dispensing pearls of wisdom from an unexpected quarter? Regarding your actual question, it sounds like you're specifically looking for stories where two things are true: 1) the jester is "speaking truth to power," and 2) the monarch learns about or changes their mind about something important. There are plenty of examples where one or the other is true, but after doing a bit of digging, I haven't yet been able to find a clear example where both are true.
There are many anecdotal examples of jesters successfully convincing a monarch to hire or reward them, and/or not to punish them, on the basis of pure comedy. Here are two from Fools Are Everywhere: The Court Jester Around the World by Beatrice K. Otto:
Tenali Rama, one of the three superstar jesters of India, is said to have earned his position as jester by making King Krsnadevaraya laugh. According to one story, he contrived for the king's guru to carry him around on his shoulders within sight of the king. Outraged at the humiliation of his holy man, the king sent some guards out to beat the man riding on the guru's shoulders. Tenali Rama, smelling impending danger, jumped down and begged forgiveness of the guru, insisting that to make amends he should carry him on his own shoulders. The guru agreed, and when the guards arrived the guru was duly beaten. The king found the trick amusing enough to appoint Tenali Rama his jester.
...
On one occasion [Emperor Zhuangzong of China] was fooling around with a group of entertainers, and "looking all around him he said, 'Son of Heaven Li! Where's Son of Heaven Li?'—referring to himself. [His jester] Newly Polished Mirror rushed forward and slapped him across the face. Zhaungzong turned ashen, and the courtiers were all terrified. The crowd of entertainers, also panic-stricken, seized the jester and asked, 'Just what d'you think you're playing at, slapping the Son of Heaven in the face?' Newly Polished Mirror, feigning innocence, explained, 'But there is only one Son of Heaven Li, so who was asking for him?' Thereupon the courtiers all laughed. Zhuangzong was delighted and rewarded Newly Polished generously."
There are also many examples where jesters are alleged to have spoken truth (of a political nature) to power in a fairly surprising or shocking way, but not necessarily effecting any change in their monarch. In some particularly bold cases, I would wonder if these were popular jokes that were attributed to brazen jesters, but never actually spoken by them. Here's one from from the 19th-century History of Court Fools, by John Doran, about Charles the Bold's jester, Le Glorieux:
Le Glorieux was a facetious fellow, and as fearless as facetious. His master, Duke Charles, used to compare himself with Hannibal. After the overthrow at Granson, Duke and fool were galloping in search of safety, with many others. The Duke was in gloomy wrath, Le Glorieux was full of wicked gaiety. "Uncle," cried he to Charles, "this is the prettiest way of being like Hannibal that I ever saw."
And another from Fools Are Everywhere, attributed to Will Somers, fool to Henry VIII:
His Majesty after some discourse growing into some good liking of him, said; fellow, wilt thou be my fool? who answered him again, that he had rather be his own father's still, then the king asking him why? he told him again, that his father had got him a fool for himself, (having but one wife) and no body could justly claim him from him: now you have had so many wives, and still living in hope to have more, why, of some one of them, cannot you get a fool as he did? and so you shall be sure to have a fool of your own.
Additional sources:
Vicki K. Janik (ed.), Fools and Jesters in Literature, Art, and History: A Bio-bibliographical Sourcebook