I have angels in my story (only two since their people were destroyed in a war right before the beginning). They are not immortal but age or illness don't affect them. They can be killed, although they are harder to kill than a human and better at healing. I have a bit of trouble naming one of them. I looked through angel name lists but I don't want to use something common, and less-known names are usually long and complicated. And they are not Biblical angels.
Why not something like Reginald which can be shortened to Reg or Reggie or even just an old name which can have a cool and modern nickname.
If they aren't Biblical angels, name them however you want. Be creative. Hell, maybe they all have Russian-sounding names.
Name them however you want. Make them no different than the kind of name you'd give any other character. You can have fun that characters notice that an angel is given such a casual name.
Reg? It would be hilarious. Every time I hear this name I think of a) the leader of the Judean People Front from Life of Brian b) Reg Shoe the zombie, of Ankh-Morpork. I would be tempted but it's a serious story and knowing myself I'd turn it into comedy...
This seems like a good idea: you can even use biblical names like this. E.g. Gabriel shortened to 'Gabe', Michael shortened to 'Mike', Raphael shortened to 'Rafe' and so on. Additionally, if you want serious names, look up Greek, Aramaic (the language of the New Testament) or Hebrew (the language of the Torah and the Old Testament) names in a baby name database. Should find something suitably exotic or grandiose.
Do you want names to be shortened? Or do you want non-traditional names? I would try and do the baby name finder, but search by "meaning" or "nationality." That's usually how I come up with my names. And only I know that it has some symbolic significance.
Want a rare one, you could use some of the ones used for the Japanese version of angels, or 'gods of death' (shinigami) Ryuk Sidoh Rem Rinne Meroko I always loved Castiel and Bartleby, although those two are quite popular.