Not, mind you, tales of their own adventures. Those can usually be tossed off in a line of summary, if not omitted entirely (presumed to be off-stage), and even when needed -- as J. R. R. Tolkien needed Legolas and Gimli to tell of the Paths of the Dead, because the alternative was remove surprise or go backwards in time -- it's all part of the plot. Sometimes, on the other hand, they have to tell other tales.
In my stories, it tends to be fairy tales.
Reading a lot of them does give you the advantage that you don't have to grab one of the handful of well-known ones. Both for the novelty factor and to actually have something that resonates with the theme, and keeps the story from falling into parts. You could invent your own, but there's always the danger of circling 'round the same motifs -- and besides, many are unconvincing as fairy tales -- at least to me after reading so many. . .
But even after reading hundreds, it can be interesting to track down the exact one that reflects father-and-son animosity, or that features something being turned into a man from the inanimate.
And then you have to decide whether the character will be allowed to tell it all, and how much goes in summary, and how well the character tells it -- the Grimms, and most other collections offered to the general reading population, clean up the tales quite a bit if only to make the story better structured.
Stuff to wrestle with.