I am not sure whether you are asking whether a whole separate organization can be created with the same name or whether you merely want to have separate groups manage the sites separately.  You could get the separate management simply by having two separate committees of the original corporation given the authority to manage the two separate sites.  You would probably want some overlap in the composition of the committees to assure that your programs are coordinated.

If you want to create two separate corporations, you probably won’t be able to create a new corporation that has exactly the same name as the existing corporation if you try to do it in the same state, but you don’t really want to create the confusion that would result if you did try to do it.  It would be better to use two related, but distinctly different names, like X County Battlefield Association and X County Museum Association.  You would have a lot of latitude here.

If you do set up two separate corporations, you may want to have one serve as a supporting organization for the other.  (See Ready Reference Page: “Supporting Organizations Qualify as Public Charities”)  It can be difficult to maintain public charity status for a small historical association and would be doubly hard to qualify two.  If one is a supporting organization, it will not have to worry about its sources of income, or even if it has any income.  All of the donations to support the combined effort could go to the parent “supported” organization (which you would want to be managed to assure its public charity status) and then be transferred as necessary to the supporting organization for maintenance of its property and program.