You say the names really need to be on a standard rotation. I'm not convinced. The situation you're describing is a very specific case, and while I sympathize with it, the obvious solution to reconstructing a lost game for a let's play is to back up your save file. Or to just say you had to reconstruct the game since you died and didn't have a backup, so you picked a new name; I've seen at least one other LCS Let's Play that did exactly that.
The given names aren't designed to be chosen specifically. If they were, I'd have just let you type it in rather than re-rolling on the name table. There's no technical reason that I couldn't let you do that. The reason it isn't that way is that it would strip away the whole point of having a given name on your character in the first place. Your given name is something you have little control over. It's not like your code name, which you pick yourself.
Iterating through the names in order would gunk up the random name generator by giving it predictable results and degrade the experience for repeat players. A more implementation-intensive solution to the problem you're describing that avoids this issue would be to have a list you can page through, like Dwarf Fortress does, and retain the randomizer as an additional feature. But this again runs contrary to the purpose of the given name.