(Preface: I'm not well-versed in language acquisition, so the context I've provided here will be dumbed-down, or perhaps just dumb.)
Context: When children acquire their first language, they typically go through a phase where they take grammatical rules (e.g. "form past tense by adding -ed to a verb") and apply them in situations where they aren't applicable (e.g. using *goed as the past tense of go). This is called overgeneralization. Children acquiring English as a first language overgeneralize verb conjugations all over the place - "I ated the candy", "She telled me to do it", etc.
Since Japanese has fewer irregular conjugations than English, I expect overgeneralization (in the context of conjugation) to be less visible, but I doubt that it is absent altogether. I have, for example, heard *iikunai as the overgeneralized negative of ii "good" in place of yokunai. (Albeit this was a child character in an anime, not a real child, so this may not be representative.)
Question: Do children make similar errors with irregular verbs and other conjugatable words? For example, might they use *shiru as the 終止形 of suru "to do" or *kinai as the negative of kuru "to come"? What about with irregular honorifics like irassharu, gozaru? (I guess children wouldn't use these as much, since keigo is mastered relatively late...)
I would also be interested in cases where children incorrectly treat non-conjugatable words as conjugatable, e.g. (to use some examples from the comments), *ippakunai as the negative of ippai "a lot; much" or *kireikunai as the negative of kirei "pretty".