at SWA it is the captain's decision.
a good takeoff brief isn't that complicated.
after 80 knots before V1, only abort for fire, engine failure, windshear, aircraft is unsafe or unable to fly
the last caveat is where the captain's decision comes in, the FO should give a clear and concise description of what they think is going on. if it isn't fire, engine failure or windshear, go flying.
that catch all is for something that might not generate a light or warning but the captain decides the plane won't fly.
that is obviously the scary one that gets people killed both ways (for aborting and for continuing).
for instance, blow a tire and try to abort, you very likely won't meet your performance numbers (poor braking with a blown tire) so on a critical runway you may be going off the end if you do a highspeed abort with a blown tire. blown tires also are a tough one if you haven't had one before in that plane, might seem like much more than it is, engine failure maybe (loud noise, swerve). there is also the question as to how much damage it did to hydraulic lines, etc. then again, if you're leaking fluid, another reason you might not stop on the paved surface.
or, flaps weren't set and you didn't get a warning (madrid, possibly) some planes won't takeoff with the flaps up and some captains might get a 'this isn't right' feeling during acceleration and abort, although flaps up is an insidious and scary one. which is why they make those warnings. at SWA we check the takeoff warning both at the gate before every flight (to get it) and just prior to takeoff (to make sure we don't get it).
anyway. I've flown both ways (in the navy anyone in the cockpit could call abort) and prefer the CA makes the decision. less debate at V1. although, any captain needs to very seriously consider the input of the FO if the FO just has time to yell abort.