Well, you could simply let them pick whatever you want.
I know that sounds amazingly powerful, but hear me out.
There's some benefit from being in a powerful Blood House, but there's also pressures on them as well (though you probably want to keep Kerensky off limits).
An in order to get a Bloodname they're going to have to perform well. Then someone with their Bloodname is going to have to die. Then they'll have to have performed well enough that they get nominated (Or be in a position to enter the Grand Melee). And then they'll have to win 5 fights to get the Bloodname itself. So getting the Bloodname that they've chosen is hardly guaranteed.
So the toughness of winning one, combined with the lack of benefits when they don't have one (for most Bloodnames), I could easily see the cost being zero. Or you could simply let them keep the Bloodname trait at the 100 that they get, so they're atleast paying something.