Rav Hirsch on 24:10 states that one reason for the doubt is that he was born before the revelation. This would make it a case in which a woman converts after she has given birth so that the child is not Jewish.
there is still a doubt whether this rule applies to cases where the child was born before the mother had received the Torah on Sinai
Technically, every individual at Har Sinai "converted" as a result of the Revelation. Thus, the statement that he "converted" could mean that he also accepted the revelation to "convert" at that time. This also makes the reference explicitly to the argument that he was not part of the tribe of the mother which would not have been affected by the revelation.
Note that the normal conversion case means that the convert is like a "newborn" and is not technically related to anyone (even his immediate relatives who also converted). However, at Har Sinai, Hashem explicitly informed everyone that their previous relationships were reinstated ("return to your tents").
כו לֵךְ, אֱמֹר לָהֶם: שׁוּבוּ לָכֶם, לְאָהֳלֵיכֶם
26 Go say to them: Return ye to your tents.
Otherwise husbands and wives would have to be remarried and no-one would have belonged to his or her tribe. In this case, the relationship that was restored was the case of not belonging to his mother's tribe.
ראבד says that the reference is to the father who converted as part of the Eirev Rav after he was born. Thus the problem was his claim to be a member of the tribe of his mother, which was rejected.
Note that this assumes that the father was not the Egyptian that Moshe killed before fleeing Egypt.