djkool14 is almost right. In an ideal elevator ignoring all the various things that upset this problem, it would do as he says (initially).

However, in a real elevator, first of all, the elevator can’t possibly freefall in an elevator shaft even if all the safety equipment fails. There’s a mass of air beneath it that becomes pressurized and cushions its fall, and there’s friction on the sides. If it were stated in the problem to ignore these effects, that would be a different matter.

Therefore the balloon rises, just because there’s still significant gravity felt by the occupants, including the air in the elevator.

He’s right in the sense that the air in the elevator will redistribute itself because the gravity is lower during the pseudo-freefall. That will help push the balloon up very slightly (the density difference across 10 feet is really very small, though, so this is likely lost in the noise… and is only relevant to the “ideal, ignore all the realities” version of the problem).

Even if it were a complete freefall, though, the balloon would still continue to rise, probably to the ceiling, because the effect djkool14 mentions will get it started moving, and inertia will keep it moving until air friction brings it to a halt. Of course, it’s impossible to predict exactly what will happen because convection is a chaotic process, and the movement of the screaming and thrashing passengers is likely to screw up the perfect ideal motions. It will wander about quite a bit.

To summarize: In a real elevator, it rises due to there still being some gravity creating bouyancy. In an ideal elevator that is allowed to come to a steady state where the air isn’t moving around, and the balloon has come to a stop from it’s initial movement upwards, it will come to a rest (eventually… but it’s almost impossible to predict when and where that will be).