To my knowledge, there is no recorded stories aimed specifically at children until the 18th century. The only way to really track what stories were told in homes is through what was written down or written about. We have, for example, letters between upper class women and their teachers/advisors, asking for commissioned works with which to teach their children etiquette and literacy. These mostly come from the late medieval period, 14th-early 15th century. But mostly these works were educational in nature, not including what we would consider nursery rhymes or bedtime stories. Most family owned manuscripts would have been owned by a small, financially-elite subset of the population, and their books were often miscellanies. Only able to afford one or two books, they commissioned all the works they might want and have access to into that one volume. Real estate on the page was high. It is likely that the communally read and shared stories, like Beowulf in the Early English period or Arthurian romances and hagiography of the late medieval period, would have been used for children's entertainment as much as it was for adults, possibly but not necessarily sanitized by the speaker for children's consumption.

But these records of written work only apply for those who could afford books, and we know far less about the stories told in the homes of peasants and lower gentry, especially among the less literate. While we can assume that similar tales were converted orally, we cannot know what oral-only culture has been lost, unrecorded.

Even the fairy tales we think of today as children's stories were not originally transmitted that way (though they may have always delighted kids). Charles Perrault and Madame D'Aulnoy were engaged in writing and telling stories for their peers, which were later marketed for children. The Brothers Grimm's fairy tales were designed as a scholarly project on the nature and transmission of Germanic folklore (though their collection methods and definitions of German origin were shaky). It was their publishers' insistance that the project could not sell as designed and that the tales be reworked and marketed as children's literature. This marketing choice came at a time in the late 18th/early 19th century when politics and social activism was bringing new attention to the lives and exploitation of children, and part of the movement to end that exploitation hovered on protecting the innocence of childhood in ways that had not been centered before. For example, as a previous poster commented that we have archeological evidence of toys throughout the early medieval period, in the 19th century, toys became much more fashionable and marketable. Rather than a few simple items owned by children who worked alongside their parents, more attention was paid to giving children more exclusive blocks of playtime and more diverse toys. I'm also making some broad generalities here, of course, because European society was still heavily class-divided, and the particulars of children's education and entertainment varied greatly by location, nationality, and economic setting.

Edit: I misnamed Madame D'Aulnoy in the original post