In the civil war relic digging-&-collecting community, that type of button sells for $1 to $2, because so many relic-diggers find dozens of them a year. That being said, the gold gilt has mostly or entirely worn off of the majority we dig. So, one with as much gold remaining as yours has might bring $5.
Relic-diggers have a menu of button-cleaning options to choose from ...everything from lemon juice to TopJob to peroxide to household ammonia (the non-sudsy kind). If a button isn't too heavily corroded, I prefer ammonia. (By the way, ammonia won't hurt the gold on your button, nor remove any gold.) I pour some ammonia into a milk-jug top, and soak the button in it for 120 seconds. Remove the button, dip the tip of an old toothbrush into the ammonia, scrub the button gently for 20 seconds, then rinse the button thoroughly under running water. Then pat it dry with a towel. See if that's enough cleaning, or if it needs another 60 seconds in the ammonia (with a little more toothbrush scrubbing).
"Straight" ammonia is a potent chemical, unsuitable for significantly corroded buttons. Therefore, some people prefer a milder version, such as ammonia diluted 50/50 with water... or instead use TopJob, or lemon-jiuce, or "warm" peroxide. With those alternatives, you have to let the button soak for a longer time. Remember, you've got to ALWAYS rinse the button really thoroughly after using any cleaner on it.
Other posters here at TN will probably contribute their own cleaning method suggestions ...especially if you ask them to do so, in a separate post titled something like "How do you clean dug brass buttons?" (Note that I specified "brass", because Pewter buttons require a different type of cleaning ...especially if they are corroded/brittle.)
