Don't bother replacing the FET with an identical one, as it'll most likely fail in the same way. The default STP55NF06L's are not the best choice, and there are better components available.
Drop in replacements with better characteristics (eg: lower Rds(on) value, suitable Vgs, etc) that shouldn't fail with the higher voltages/more current you're possibly pulling:
IRLB8743
IRLB3813
FDP8870
AOT240L
CSD18502KCS
Many of these you can pick up from Element 14, RS or Digikey.
Note: IRLB models also come in a similar package called IRLU. Same sort of mounting but not as good for mounting a heatsink. Unless you're drawing some serious current however, both the above will stay cool without a heatsink, though you should make sure there is moving air around them (eg: a fan).
PS: If you use a 12V heatbed at 24V, you will pull twice the current (so >20A), and put out 4 times the power (bed will heat too fast, which can cause weird oscillations in temp as it constantly overshoots the temperature). The tracks and connectors on RAMPS are not rated for 20+A (try about 12A absolute max, and less on cheaper versions), If you're having issues with the heated bed reaching max temp, try some marginal voltage increase instead like 13.8V or 15V, rather than 24V.