For me it is
1) Playability and comfort. Specially the neck radius -SR type over P- and high frets reachability.
This body probably would be an example of a big no for me. If BTBs had a narrower neck and string spacing they'd be perfect for me as I love their bodies. Weight is not much of an issue as I can't play standing up. My thighs could get used to less weight, though I haven't reached that point yet.
2) Tone. Darker, warmer and thicker over bright and thin. It will depend on whether I'm looking at a expensive bass or not whether this will be more or less important. If I'll be buying a $1000+ bass, I want it to be as close to perfect as it is as it possibly can. I mean who would want to spend money on pickups and electronics on an expensive bass.
3) Looks if two or more basses are tied. I want to be able to watch it resting in its stand when it's not in my hands.
I'll take a natural finish or stained visible grains over solid colors a la P and Js any day for example.