Bottles with that odd round deposit on the base are what we call iron pontil. That was a form of manufacture used from roughly the 1840s to a little after the civil war. Basically the glass blower had a rod with a fairly big round ball on the end. When he was done with the blow, he'd attach that rod to the base of the bottle to hold it while he finished the lip.
So that bottle was roughly 100 years old when your grandmother dug it up. It's in great shape for a dug bottle, particularly for a soda. Those often saw repeated usage and are in rough shape. That one looks really nice.
That is indeed a very nice bottle. I think that Iron Pontil sodas in that very light aqua color are very cool.
They are probably more rare than the green one. But not being a colored bottle they are likely less desirable.
But I also wanted to address the comments about when Iron Pontil bottles were in use. From my observations,
In the Phila/S.Jersey region, They stopped using any form of pontil around 1857. I believe that this was a wholesale change in the bottle making industry. And likely swept across the country from east to west in a short period of time.
This improved way of producing bottles would have been advantageous for any glass maker. You certainly wouldn't
have wanted to be producing bottles with the unsightly Iron Pontil scar when your competitors were not.
So as far as I know, all pontiled bottles were produced before the Civil War by a few years.