There's a service manual for the synare III online, but it's a bit terse - It only shows schematics in fragments, and doesn't show everything.
the synare has an auto power-off circuit to save battery life if it's not being hit. This is basically a "oneshot" being fed by the trigger signal... It seems like this power system feeds the amplifier sections in the drum synth, which would pull current from the batteries all the time otherwise..
so... to me, it sounds like your system isn't powering on so neatly when you don't hit it for a while, and this is creating a noise..
I agree with erroraudio that it's probably some bad caps somewhere. (in front of the regulators that get switched on when the oneshot fires, or maybe even in or around the oneshot - making its control signal oscillate a little bit, when turning on, rather than go straight on)
they're a very simple design with very few components, and are easy to get into. It would be simple and cheap to have all electrolytic and tantalum caps replaced, if you can't do it yourself. This would probably fix it (but no guarantee...)