Interestingly, in English and other languages, we treat "Adam" as the private name, not as a noun, but in the Torah it is a noun, for example (Ber 2.25):
וַיִּהְיוּ שְׁנֵיהֶם עֲרוּמִּים הָאָדָם וְאִשְׁתּוֹ וְלֹא יִתְבֹּשָׁשׁוּ׃
The two of them were naked, the man and his wife, yet they felt no shame.
So, did Adam have a private name at all?
Interestingly we saw the same difference in calling his wife's names: in the beginning, she was called "wife" as the Posuk says "ייקרא אישה" but later he calls he the name Chavah - "וַיִּקְרָא הָאָדָם שֵׁם אִשְׁתּוֹ חַוָּה" - specifically "שם". But Adam wasn't given a name.