An I2C backpack for the 1602 reduces the effort to connect it all up.
A potentiometer (say 270 degree sweep - linear) would probably give a maximum granularity of 120 minutes and it then becomes difficult to select an exact minute.
If you want further precision, you'll probably have to use a (the) button to switch between "set hours" mode and "set minutes" mode in order to set hours and minutes separately.