For me, provably fairness is a must. On top of that, reputation, house edge % and whether the games are fun are the three most important factor for me.
Provably fair is not a must, it is much more than that - it is industry standard. No one will gonna take seriously gambling site without this system implemented.
But there are cases when you simply can't have Probably Fair system in place - for example PF system won't work with live games like Roulette or Black Jack.