How to choose the right MOSFET for this circuit
Two main issues: Provided you give your MOSFET enough voltage at the gate to turn it fully on, you reach the lowest possible resistance of your type of MOSFET and allow it to work with as little losses as possible.


First, your MOSFET must be able to work with the "on" and "off" logic
levels provided by IC1. "Off" isn't critical, you want to take care
for "on". Look for the minimum HI output voltage provided by IC1 in
its data sheet. The value is likely called VOH,min. This
value must be higher than the MOSFETs maximum threshold voltage
VTH,max. Allow about 0.5 V of safety margin between the
two parameters. This way, when your IC outputs a high voltage, your
MOSFET is guaranteed to fully turn on.
Second, The MOSFET must be able to handle the current required by the
load. Find out what current ILoadyour module (LCD1)
requires on the pin connected to the MOSFET. Check the MOSFETs on
resistance RDS,on. Calculate the loss dissipated by the
MOSFET using PMOSFET = ILoad2 *
RDS,on and check if the MOSFET package can handle it (cf.
thermal data in the MOSFET's data sheet).