Looking only at info. from the NTSB, there are some interesting points:
A directional control problem became evident at high speed on the TO roll; corrective action using rudder was unsuccessful, as was a subsequent attempt at using the tiller and the TO was rejected. The order of these actions (as in when the RTO was commanded) is not completely clear as there is some semantic confusion in the initial bulletins but they seem to imply that the RTO was intiated as/after the aircraft left the runway. This makes the MLG & NW tyre marks on the runway (as reported) look rather odd.
To get rubber left behind you need a) a lockup, which can be discounted as there were no flat spots found on the tyres or b) ABS working at maximum effect, which would imply the RTO came much earlier than stated or c) a large enough slip angle to abrade the tyre. Going with the last option for a moment, that means the aircraft's heading had diverged *significantly* from its track. Something happened, either aerodynamically or mechanically (or both) to introduce yaw to the stage that it was 'drifting' along the paved surface; the forces involved eventually took it off the side.
Examining purely environmental effects, it would take a fierce, prolonged 'gust' to cause this loss of control, given the type of aircraft and a dry runway. Also factor in the speed (>100kts), the veering of the wind towards the runway alignment during a gust and the increasing rudder effectiveness and it seems unlikely (but still a possibility, of course). This is with the primary flight controls functioning normally. There's not much in the METAR/TAF at the time to suggest anything unusual: no CB activity or precipitation/virga; no LLWAS info. and -4/-14 with cloud at 4000'AGL. The crosswind was a known issue.
As far as the (brief) use of the tiller goes, I'd expect that to be along the lines of: 'We're going to crash, try anything that might help!' as to use the tiller you have to let go of something else, i.e. the control column, which is something I'd not want to do close to getting airborne in a crosswind. (I don't know CO SOPs but I assume it'd be P1 hand on TLs up to V1.)
Pure speculation: In a fight, which would win - nosewheel or rudder? Low speed I'd give it to the nosewheel. >100kts, I'd say rudder. Points towards some sort of rudder malfunction, doesn't it? Add in a brisk crosswind... Who knows?