Hi.
What is the most effective way to switch a battery power supply with minimum voltage drop, when I don't care about speed, just maximum effectivity and smallest size?
Relay
Relays are unfornately too large for my project. I tried using transistors, but they drop their voltage too much. Is there anything else I could use?
Oh, use a MOSFET with low RDS(On)
Oh, thanks. Could you maybe tell me an example (MOSFET model number) that would work for 3.7V 2A?
How much looking have you done? There are relays that will switch your current that are about the same size are a MOSFET you are looking for.
Are those small relays really as effective as a mosfet?
Here are some logic level MOSFETs that fit the bill.
An AO3401 (P Channel, -2A, Vgs = 2.5V @ 60 mΩ) is good.
Never had a low current (less than 5A contacts ) relay fail in many many many years with working with them 
BTW
A very good N channel MOSFET is the AO3416 (5A, 21mΩ at Vgs = 1.8V).