I think I may have identified the most obscure of obscure Trek props.
After a bit of searching on the Trekcore.com photo archives, I could only find it in a couple episodes and I still can't even think of a name for it.
It's not the exact model but it's the one I had laying around.
Here are the only other spots I can find this goofy prop.
With a little TNG style almond spray paint, I'll be able to make props with a prop. :)/>