Can a surge protector prevent nuisance AFCI breaker trips?
The job of AFCI is to protect the house from series or shorting arc faults, which cause fires. 

The main reason AFCI breakers are required today is because so many houses are built with backstab connections to switches and receptacles, and those so often fail in a series arc fault mode.   

Keep in mind, the builder's electrician is the one who wired the place.  So the last thing he's going to say is "it's tripping because I used backstab connections when wiring your house because I was in a hurry, and one is failing as they often do.  Sure, I'll do a go-back and spend two hours taking apart all your receptacles and moving the wires to the screws, or fitting the $3 receptacle instead of the 60 cent one". 

That said, it can also be a failing appliance.  However if you move the appliance to a different AFCI protected circuit, this circuit will stop tripping and that circuit will start.