The FO:NV wiki (warning: huge spoilers) has the relevant info, although it takes some work to piece together.
Mr House: your only option is to destroy the Brotherhood. There is an option to convince him to ignore the Brotherhood in the game script files, but it's not accessible in game.
Yes Man: the Brotherhood will never support you directly. They are, however, willing to ally with the NCR, if Elder McNamara is still in power - which is good for you, too, if you're on good terms with the NCR. Even if you're playing the Yes Man questline, you can still get this to happen by following the NCR quests until you reach For the Republic, part 2. Convince the NCR and the Brotherhood to ally and then report back to Yes Man.
If you're following one of the other main quest lines, Caesar gives you the same options as Mr House, and the NCR the same options as Yes Man. (Strictly speaking, it's Yes Man who gives you the same options as the NCR, since that part of Side Bets piggybacks off the related NCR quest.)
The ending will vary slightly depending on which of these paths you take; according to the wiki, there are five variations, depending on whether the Brotherhood was destroyed, and, if not, whether or not you got the Brotherhood and the NCR to sign a treaty, and which of the factions (if any) you sided with.