Anything that's rubber fails. Primarily, coolant hoses and belts. Also fuel hoses and hydraulic hoses.
Cooling problems are probably the #1 failure.
If you're running a high-mileage engine, it's probably wise to put a new set of bearings in it, even if you don't do any machine work or replace anything else, at least you'll have fresh bearings. Cheap insurance, plus it gets you inside the engine where you'll spot other potential problems. Ditto with head gaskets if it's a really old engine or an engine prone to head gasket failure.
Brakes, as Rob mentioned. Just put a set of race pads on the car with high temp fluid. Don't ask questions, don't wonder if generic parts store pads will do. Just do it. Add some brake ducting, too. Why not? Just do it.
Electrical systems fail. The more complex they are, the harder they fail. Unless you're working with something older than mid-80's, you're going to have to keep the ECU and a lot of the factory engine harness. That's fine. But, don't rely on factory wiring for anything else. When it comes time to troubleshoot electrical problems, your simplified "wire from the kill switch through a fuse to a switch that goes directly to the fuel pump" is gonna be a helluva lot easier to figure out than whatever convoluted ECU-controlled, safety-interrupted, relay-switched system the factory gave you! Same goes for lights, brake lights, etc.
Gauges are not considered in the budget. Install some in a visible location and be sure your drivers know how to read them. A giant "low oil pressure" light could save your engine.
Exhaust system. Another "safety" item. Spend the $200 and do it up right. This can save you black flags and hours of off-track repair time.
Replace the clutch. Find room for an eBay clutch kit in your budget. Just do it. Like so many other Lemons teams before you, you'll kick yourself for the clutch that failed during the race. Just do it.
A lot of the wheel/lug failures that Rob mentions are mechanic error problems. Invest in a torque wrench. Use it. Undertorqueing lugs will result in failure, as will overtorquing. Use the torque wrench on wheel lugs, major suspension components, brake caliper mounting bolts, etc.
Whatever you overlook will fail. Whatever you knew was "iffy" and didn't replace or repair will fail. Take the hit at BS judging if you have to. The 10 lap penalty they'll give you for the obvious new radiator will be better than 2 hours of down time scrounging a new one, or an hour replacing with the spare you brought... or the trashed engine due to allowing it to overheat.
Lemons South 2008 - Fail, Lemons South Spring 2009 - Fail, Lemons Detroit(ish) 2008 - Fail, Lemons South Fall 2009 - Fail, Lamest Day 2009 - Fail, Miami 2010 (Chump) - 2nd!, Sebring 2010 (Chump) - Fail, Cuba 2010 - Crew Chief, Roebling 2011 (Chump) - 8th!, Sebring 2011(Chump) - 19th!