I have land an minerals in Texas & I am not familiar with Kentucky. Here is what I have learned about leasing mineral on the Texas Gulf Coast, southwest of Houston, Texas.
If your land is in a shale play, the signing bonus can be in the thousand dollar range.
If not, $150-$400ac has been the acceptable range on the Texas Gulf Coast where I have land and minerals. It depends upon whether it will be a "wildcat well" (a discovery well in an untested area) or if it is in a field with proven production. In Texas, the Railroad Commission regulates the oil & gas industry & gives out the well permits. Find out what government entity does this in Kentucky. From that entity you can find out if any permits have been given in your area or if there any active wells.The may be able to give you the names of oil companies in the area of your minerals. Call one or two companies and ask to speak with a land man. Ask him what he considers a respectable lease offer in your area.
Most important is your royalty percentage. If it is a big company or if the area is "hot" (active), you may be able to get 1/4 royalty. Start out asking for that. You may have to compromise with a 1/5. Never accept less that 1/8. Your success in negotiation will partly depend on how much of the mineral acreage you own - or that your family owns. The family members should "stick together" for more leverage. You and your brothers do not have much of the minerals; so, you may not get more than 1/8 royalty, but you can ask. Ask your aunt & uncle what offer they received and if they accepted it or not. If your family also owns the surface, you have a little more leverage. If you own the surface under the drill site, make sure you are paid damages for the drill site.
Ask for $25ac or $50ac delay rental. The signing bonus may be less, if the company pays delay rentals. A "paid-up lease" does not have delay rental payments. If the lease/offer provides for annual delay rental payments, the company will lose the lease if the payment does not arrive on or before the date of the lease. If the company drills and gets production, the company will not be paying any delay rentals. It will hold the lease "by production." Ask the land man when the company plans to drill and when they can get a rig. Ask them how long the company has been in business. Look the company up on the Internet. Try to find out if they are reputable and how many successful wells they have drilled in the last 5 years.
"Eminent domain" is not the proper term. That term is used if a company is putting a big pipeline in that will carry gas to a gas plant or water to a city. It is a taking of land for public use if you do not sell at a reasonable price. It requires a lawsuit. But, if a company leases a majority of the mineral interests, the company can drill without leasing all the mineral interests. They are more apt to do that for a seismic permit. They really prefer not to do that for a well because after their drilling expenses are met, the unsigned mineral owner gets 100% of his/her royalty interest (not 1/8). It may take 2-3 years for the company to recoup its expenses. If the well plays out before the company's expenses are met, an unleased mineral owner will receive nothing. The well may be a "dry hole" and you will not get anything. So, it's risky for the mineral owner.
If the family has enough minerals to justify the cost of an oil & gas attorney, I recommend you use one in Kentucky. If you prefer, he/she can negotiate for you. He/she can also provide your lease or review the oil company's lease and add an addendum of provisions to protect your rights. Find out what the attorney will charge and see if it is worth it.