OK . I'm starting to zero in on the piece. More questions:
1. Is it a real narrow u-channel-like piece that installs onto the door edge?
2. Or the fairly fat and thick piece that sits on top of the inner door (where the door locks would be in on most cars). That's where I rest my arm when I drive.
Piece 1. or 2. ..... or none of the above.
Do you have a Factory Shop Manual?
If not GET ONE. Helms has them, Google Helms, or go to ebay.
It's indispensable for working on your Vette, and just reading through it will teach you ordinary stuff about just driving you car that you never knew. It's also helpful to get both the Chiltons, and the Haynes too. Between all three of them you get an assortment of pics, views, helpful hints and the like that reinforce and support one another.
Of course this BB is also the best, and there are often references to articles in Corvette Fever that are great step by step with pics.
Let me know re the mystery part. A pic is nice if you have a digicam.