Is it a wood snare, or metal? What type of strainer is it? Quite often if its a buzz problem it has to do with how the snare wires are held in place, or how the bed is cut. Make sure the bed is centered well and isn't twisted so that one side is looser than the other. Adjust the snare wires or straps so that the bed is centered evenly.
A well cut snare bed is crucial on a wood snare. I always have the bed re-cut to a deeper level. That way the bed can rest on a slight curvature of the head. If you do this, you will hardley need any tension at all.
If the snare has one wire that is streched, nothing you do will fix the rattle! You will need to replace the snares. Run your finger across the snares while it is tensioned and see of one or more rattle. If it does, buy a new snare bed.
If all is well with the snare bed, then the problem is with the tuning of the rest of the kit and you have a frequency that is reacting with the tuning of the snare.
Another little trick for snare buzz is to tighten the lugs that correspond to the crossing of the bed (center of the head), and then detune the ones closest to the ends of the bed. This has worked wonders in situations where I can't solve the rattle problem.
