Selecting a logic level MOSFET compatible with 3.7V power source
The threshold voltage is where the MOSFET is mostly off (a couple hundred uA) so you definitely want to exceed that whenever the MOSFET is supposed to be 'on'. 

It can withstand Vgs of +/-8V so you're fine on a 3.7V supply, and it's rated with 2.5V drive for Rds(on) so you're fine for drive down to below destructive voltage for the battery.