If you can get a scope on it and make it trigger on the glitch, the output of that would I think be useful for figuring this out - but that might be hard to do.
Without that - I'd try as you suggested putting a cap between power and ground, and seeing if that helped... (nothing special here, just like... 10~100uF probably? The exact value is probably not critical). If you're able to see the glitch on the scope, you could of course see what impact the cap had and assess if you needed a larger one, or whether it wasn't helping.