I have heard that some people do not allow a child to see his or her reflection in a mirror before they can speak (or say their name, possibly). Is there any traditional source for this practice?
In Be'er Moshe, R' Moshe Stern relates how it was the custom in his home not to allow children to see their reflection in the mirror before their teeth come in or before they begin to speak. In regard to several of these type of (bizarre) customs, he writes:
וכל אלו הג׳ מנהגים בכלל מנהג נשים זקינות שעליהם כתב הרשב״א שאל יזלזלו בדבריהן ובמנהגיהם כי בודאי יסודתם בהררי קודש גם אם נעלם הטעם ממנו
and these three customs are all among the custom of old women about whom the Rashba wrote that we should not belittle their words and customs because their basis is doubtless in mountains of holiness even if we do not see the reason
[However, a commenter on this article quotes R' Shmuel Kamenetzky as saying that the custom to tie a red string on a baby's carriage (one of the customs mentioned by the Be'er Moshe above) is nonsense and should not be worried about.]
It is superstition, which Jewish women likely absorbed from the surrounding culture. I wrote in another answer about upsherin and the Pennsylvania Germans / Dutch, where the Pennsylvania Germans are continuing an old superstition from Europe (and where upsherin is explicitly not practiced by Tannaim/Amoraim).
Similarly, R' Menasheh Klein's endorsement of throwing a tooth into a mousehole and saying an incantation -- a German superstition, associated with Loike.
Similarly, this. See how the Pennsylvania Dutch don't allow a child under the age of one to look into a mirror, lest it become proud. (Other similar superstitions exist prior to baptism.)
Rabbi Moshe Stern was cited in the above answer (and in Revach) saying what he says, but he is incorrect. I understand the impetus to say this, and that he has the Rashba as a basis. But the Tosefta Shabbos lists a bunch of superstitious practices, and forbids them, because Jewish women were engaged in them. Why else bother to forbid them? And where it is possible to document the parallel, it is incorrect and misguided to say that they are not superstitions and to promote them to the status of holy customs with halachic basis.
I read from Rav Mutzafi that there is no problem for a child to see his reflection in a mirror. I believe he said that those who discourage a child from seeing his reflection have no source for this "custom".