I believe that if you desire to gain property on the Moon as a U.S Citizen by claim then you would have to follow the http://www.legendsofamerica.com/AH-Homestead.html"
If you Gain Property other than just by Claim you will have to purchase it by those who have set their Flag or Marker upon it.
Also the Moon Property would have to be Marked out with Property Boundries accourding to the Federal Goverment Laws and Regulations and National Laws to make it legit or your Property Claim is meaningless.
You also may have to pay taxes on the Moon Land.
If you are a Billionair then you might achieve a Moon Property Claim and be able to keep it.
However, I would think the Moon will be nothing More than a Space Base Station/Science Lab in the Future
because once the price of setting up a Moon Base was Established it would Pay for itself and would begin to make a Net Profit from Satellite Owners/Clients because it would be cheaper to launch Satellites into Space from the Moon to Earth Orbit.
The Moon would require a Defense System to keep People safe from Space Particals, A location on the Moon that is least effected by incoming Space Rock would seem to be Prime location for an established Moon Space Station, All Stratigies in avoiding incoming Space Material like Meteors would have to be established, Under ground Facilities would probably be required as a Shielding Addition to Meteors and Radiation Exposer, Being on the Moon during a Mega Sun Flare event is a Scary Situation indeed or even an inbound space rock weighing even one pound at 35,000 MPH coming on Radar within a Quater Mile of your Location is a very whitening event.
Even with an advanced Defence Technology it is still extremely dangerous.