Should I use a mechanical or solid state relay for a hairdryer
The relay will probably give you something like 150,000 operations life under that service. So at 3.5 minutes per cycle =  17 cycles per hour, you would get about 8800 hours operation or about 1 year 24/7 before the relay wears out. 

If that's acceptable, the relay is easy and cheap (assuming the safety agency approvals work for your jurisdiction).  

If not you can consider an SSR with a suitable heatsink for about 7W. The cyclic heating and cooling of the SSR can cause early failure, and I suspect off-brand types may be more susceptible so between that, the heat sink and the extra cost, they are not a panacea. They also almost invariably fail 'on' unless subjected to massive overcurrent, so that may not be as desirable (you should, however, make sure that the consequences of an 'on' failure are acceptable regardless of which path you choose to follow). 

The hybrid approach @jonk suggests can work nicely but it's still good to have enough heatsink available that if the relay fails to close for one reason or another the SSR is not going to fail. One approach is to put a slow-blow fuse of a few A in series with the SSR.