It really depends on how much gate voltage your MOSFET needs. If it's a "logic" level MOSFET, then 5V might be enough.

Get the curves of your MOSFET, and see if it can handle the solenoid current at the gate voltage you plan to use.

Alternatively, if you're only going to build one of these, just measure the Drain voltage on the MOSFET when it's on. It should be well below 1V.