There are latching relays that only require a pulse on their coils to operate, reducing the power wastefulness. That is, assuming they are not operated at a very high rate. Also, there are small, sensitive relays with coils that consume little power.
There are solid state options too, but the details depend on the exact voltages and currents and the loads the contacts are seeing. Are the relays switching only pushbuttons from game controllers, for example? If so, maybe CMOS switches can be used, but the actual load current and load voltage is key info to arrive at a successful solution.