Check your transmission fluid level.
There are a few things that first come to mind. One is the throw-out bearing, but you said the noise only occurs mid-way though the clutch travel, so that almost completely eliminates that idea. It could need grease on the pivot ball in the bell housing. Not really an easy way to do that.
Does it ONLY happen when changing gears? Like, does it make noise if you are cruising along and push the clutch in (but don't shift)? How about at idle or with the car off? Or does it only make noise AS you are shifting?
If it's the latter, my bet is low trans fluid.
If it's the former, then it very well could be the pivot ball needing lubrication. That's a pretty common issue with push-type clutches. To do it, you'd need to drop the transmission (that's the easy way), or snake a grease gun with a long hose on it through the inspection port and dump some kind of thick grease behind the pivot (that's the hard way).
120k on the stock clutch, it could also be wear. That's a LOT of life for a clutch, especially in the traffic congested areas of northern Maryland.
If you can give specific situations where the noise occurs, then that would help narrow things down. So far, those are the three best guesses I have.
You can inspect clutch life visually through the access port on top of the transmission. You'll see the pressure plate (it's silver), and the flywheel (it's got teeth on it), and then something sandwiched between them (the clutch). Inspect it's thickness.
Trans fluid, just check the dipstick with the car warm but turned off. The dipstick is on the passenger side, right near the firewall. It's yellow, and you can't miss it. It should read to the "full" line when warm and idle. You need to wipe it clean before getting a reliable reading.
There really isn't a check for the pivot ball needing lubrication, other than the fact that the noise would sound the same each time you push the clutch regardless of whether the car is moving, idling, or turned off.
