> I think the MOSFET shouldn't be getting that hot.  What MOSFET is it?

It really shouldn't be getting hot in this sort of service.

RdsON too high for this application, or not enough gate voltage to turn
it on would be my first suspicion.  Second, I'd look at the snubber on
the solenoid coil, and make sure that it's keeping the fet from
avalanching.  If the voltages you see there are anywhere near the VDS
rating, then you are probably in avalanche.
Also watch the VDG rating.

> Actually, 5V may not be enough to turn some MOSFETs "on" all the way.
> You have to check the curves on the datasheet or just choose a MOSFET
> with "logic-level" inputs.

Many don't turn fully on till around 10V. 
Simple check, what's the voltage between drain and gate when "on"?
Work that against the solenoid resistance and applied voltage, and see
what your RdsON figure is. If it's not real close to the rated number,
then it's not really ON. 




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