Hi Mike,
I spent 6 months driving round the US in 2003/4 and had a rental car for the duration. It wasn't cheap but it was less hassle than buying a car and trying to insure it for travel across various different US states then having to try and sell it. I did around 38,000m through 21 states. The maximum duration that you can rent a car in the US with most rental companies is 56 days but you can take out back to back rentals and book them in advance which is what I did. The only hassle is that you have to take the car back at the end of the rental and then take out a new car under the new rental agreement. Unless you know where you are going to be going this means taking it back to the place you rented it from originally unless you pay additional one way drop off charges.
If you are going for 3 months then a 56 day plus 35 day rental should cover you. Its often cheaper to rent in multiples of 7 days as well so 7 days will be cheaper than 6 even if you return it early for the day you don't need.
I rent cars in the US every year for 6 weeks and for around 8 years I have been using www.skycars.com which is UK internet based consolidator, they are generally very competitive and give good value for money there have only been a few occasions they have been beaten on price and if they are they will match the price, they also use most well known rental companies. They fixed me up for my 6 month US trip by booking 3 back to back 56 day rentals with Alamo. I even ended up dropping the cars off in different US states from the pick up locations and Alamo waived the one way drop off rental charges but that might not always be the case. You will need to contact them direct though phone/e-mail though as a 3 month rental is not something you can do on the internet.
If there are several drivers going it may also pay to go for the 'Gold' rental packages which give you up to 3 named drivers for a small additional cost and you get your 1st tank of fuel free so you don't have to fill the car up before you take it back. 21 is key date for most US states so as long as you are all over 21 you should be OK renting.
It will also save you a lot of hassle having your driving licence on you as photo ID as if you are 21 you will be asked for photo ID if you go out drinking - just remember don't under any circumstances drink and drive (2 light 16oz pints is about the limit if you do drink) as the Police will not mess about they will simply lock you up there and then, you will get a criminal record and that will mean you will not be able to return to the US again if you plan to return there.
Enjoy ya trip
Nige