I have a new (2 months old) Xbox. Recently (Since Black Ops 2) it occasionally has a lag, noticeable primarily on the right stick.
The intermittent part is that it either works correctly when I first turn it on or it doesn't. Sometimes I'll go several days with proper operation and then other times I have to turn the console off and wait 15-30 minutes before coming back and it will work correctly.
I have never been able to just cycle the console off and back on and have it work correctly. It only "fixes" itself if I leave it off for a time. I've tried changing batteries, re-pairing the controller with the console, cycled just the controller off/on, and I even purchased a new controller thinking there must be something wrong with the original. Nothing seems to make a difference.
Turning it off and waiting is the only thing that usually fixes it. I say usually because sometimes waiting 15-30 minutes doesn't correct the problem. I've had entire Saturday's where I've checked it occasionally and it never worked correctly then on Sunday it has worked through three or four separate uses throughout the day.
Very frustrating. I only play Black Ops 2. It's the same TV. I've tried all the TV things that everyone always suggests for controller lag but that is not what I'm experiencing.