According to Legends of Tomorrow it's...
I don't think that's going to be comic Rips' name. Maybe his middle name since Booster has his dad's name as his middle name. I don't see Booster as the type to name his son after him. While LOT Rip is supposed to be an orphan that still seems iffy to me.
I don't want Rip to be Theodore because I'd like him to have something more original as his first name but that seems more like a name Booster would give his son. Even so I think this will be one of the few nods the show gives Booster so I'm happy about that.