Relays operate on electromagnetic principle. I found a relay on Digi-Key with a coil rating of 23.3 mA @ 6V. RELAY 1A 6VDC LOW PROFILE PCB - TQ2-6V
If you want real low current draw, consider using CMOS technology 4049 or similar :
Again, please excuse the lousy drawing. I think the 10Ks can be replaced with 1Ks, but breadboard it up and see.