Being a small oval instead of round, it's not a button, it is part of a cufflink. Yours appears to be made of Sterling silver. That specific "form" of cufflink (two small ovals with a loop on their back, connected by a short piece of wire) has been found in Colonial-era sites, and a few have come from civil war sites ...although most were made of plated brass, not Sterling like yours.
I think Buckles is right,although it would of been nice to see some hallmarks on the back to be sure,maybe to early but not many metals look like that so silver is a good bet for them.Although cuff links have been around for at least the early 1700's I know & this will prove .......................... Cufflinks
with out much change in design i think yours would date late 1700's maybe even early 1800's.
I got quite a few now of them & they can be interresting.Here's a half set I found a couple weeks ago one of my best pair yet.It was neat to me because of the back attachment & the site I found it on had not produced nothing 1800's off it yet.But the back was like none I'd seen before...........well it still has not produced nothing 1800's because I finally found a back like it on this pair. UK Finds Database - - - UKDFD
Take care,
Pete,