The Doctor: A question only I could answer. A truth field to make sure I'm not lying. If I give my name, they'll know they've found the right place, and that it's safe to come through.
The Time Lords weren't only seeking the all-clear, they were looking for the correct universe. We've seen parallel universes in Doctor Who, where individuals from "our" Doctor's and his Galifrey's universe have counterparts. If one takes the licensed novels, comics, and audio plays as cannon, Galifrey itself has counterparts in the multiverse. The Time Lords were looking to return to their original universe, but didn't know if that was the one they were broadcasting to. If they got the wrong universe, and it had a Doctor who would come to investigate the question, the only way they could be sure that it was "their" Doctor is by his name. If The Doctor were free to lie, he might give a false name, sending Galifrey searching the vast multiverse for a destination they'd already passed by. The Time Lords were not in The Doctor's best graces when last they were in contact; they couldn't be sure that, given a free hand, he might not send them wandering into infinity.
This is, of course, partially conjecture, as we don't have a solid answer as to why a parallel Doctor couldn't have "our" Doctor's name. It does, however, fit with the mythos that they have built up around the Doctor's name. It seems reasonable that a Time Lord's name may represent a unique identifier down to the universe they hail from.