1. They should be willing to pay at least several hundred dollars an acre to amend the lease, maybe more. Being a business, they will offer as little as possible, but you can ask for much more.
2. Why the letter to her father? Probably 2 different departments involved. The department that pays royalties has made the change, and the department that does leases (the Land Management department) maybe is working with older records.
3. Always a good idea to have as much information as possible when working with leases and mineral rights. Do you know if your wife's father was the one to sign the leases, or someone before him? I understand the Doddridge courthouse is crowded, so the more information you have to begin with, the better off you'll be when you get to the courthouse. If you know an approximate time on the current leases, and if it was her father or maybe his relative who signed, you can go to the indexes and search under his name. The lease probably isn't to Antero but has probably been assigned to them since the original signing.
4. While there, if you can trace the deed to the mineral rights that would be good. If you know how long it has been in the family, whether a relative owned the surface, etc. you can go to that index and see what you find. It can get complicated, and if you want to do this and have questions, please ask. There are people in this group who have done this. I am not an attorney but have learned a lot about title tracing from my sister, an attorney in a different state, and from reading here and just doing it. I am not qualified to give a legal title opinion, but I know when I have enough information, or have to get more.
5. Antero should be able to give you the information about the leases involved, what lease book number and page for them, etc. but if you can, it is good to get your own information as well.
6. Before signing, it is always a good idea to have an attorney look at the paper you plan to sign. Sometimes the company tries to make other changes to the lease as well as pooling. It is to their advantage to have your wife agree to pooling, so they should be willing to pay more that $1 to get that agreement.
7. If you know where these wells are that you are being paid royalty for, and can research where the new well permits are being granted, you can see if your property is in an active location. Much of Doddridge is active now, so you probably are near that activity. And, I have read (don't know how true it is) that where oil and gas are found in shallower depths, probably there is some at deeper depths. So the fact that your property has current wells is a plus.
Good luck with this, and please tell us how you are doing and if you have any questions.