First, determine the maximum value of Rds(on) that you can accept. This is typically determined by the maximum current you will be switching, and the power dissipation in the mosfet that you can accept. Unless the current is very high, then you will normally want to design for no heatsink. In this case I suggest limiting the power dissipation to 1W for a TO-220 mosfet, or 0.5W for one in a smaller package (but check the datasheet in case it is lower). Use lower values if ventilation will be poor, e.g. in a sealed box. Since the static power dissipation is I ^ 2 * R, you have Rmax = Pmax/I^2.
Now look for mosfets which have Rds(on) no higher than that value, specified at the gate-source voltage (Vgs) you will be using or lower, and the maximum current you will be using or higher. Also the drain-source (Vds) rating needs to be high enough for your application.
When using mosfets for low speed switching applications, that's usually all there is to it. In this particular example, there is an additional consideration that Rds(on) should be low enough such that the voltage drop across the mosfet does not cause the brightness of the LEDs to decrease significantly as more LEDs are turned on.