I know Dick eventually becomes Nightwing, Jason eventually becomes the Red Hood, and as far as I know Damian becomes Batman in some places. What about Tim? I know he changes his name to Red Robin eventually, and in an alternate future he becomes Batman, but has he ever taken a different name like Dick and Jason?

If you don't count Red Robin (and I don't either) then no, he hasn't. Technically the alternate future version of Tim Drake came back at one point and briefly used the name 'The Savior' in the Superson's story arc last year. But I don't really count that either. Frankly, I've been waiting for him to have his 'Nightwing Moment' and shed the Robin moniker for his own adult identity for a while now. I really really thought that after the Tynion run on Detective Comics we were reaching the point where he was ready for it. But this Bendis Young Justice thing seems to have taken him backward, I for one am not super wild about it.