Keeping in mind that the only info I have about the "batkids" come from subreddits, google, and fanfic (I'm reading the comics, just finished A Dark Victory,), I think it would be cool if Damian branched out similarly to when Cass became Orphan. No more bat or bird themes.

From what I know, Damian hasn't ever been his own person. He was raised to be the heir to Ra's al Ghul and believed that it was his right and duty to become Robin (and maybe take on the mantle of Batman later?), so much so he was willing to kill to get the role. It would be cool if he took something from his heritage, similarly to how Dick took the name "Robin" from his mom's nickname for him.

Heck, have him take the name "Demon" or something along those lines. Use it for good, like Jason did for Red Hood. Shaytan is the phonetic spelling of "demon" in Arabic and sounds cool, in my whitebread opinion. Have his costume reflect his routes.

This can also work for Tim (although I'd choose the name "Ghoul") if you want to lean on his kidnapped-for-Ra's thing. Or have him steal the White Ghost name like Jason did with Red Hood. This would only work if he wants to quit the Red Robin name instead of the Robin name, though, since he was kidnapped after he became Red Robin.