Lindsay Ford used to offer car rental but I don't see it on their website but it would be worth e-mailing them to check. They're within walking distance of Coleraine train station
Unfortunately their rental site isn't showing - the nearest would be Ballymena but they may be able to deliver to the Coleraine car showroom?
Roadside Garage (Edwin May) is also showing rental options. Their location is not convenient though so you would either need to get a taxi (should be £3.50- £5) from the station or see if they can collect you at the station.
Thanks everyone for your speedy replies and helpful suggestions! I've emailed a few and am waiting on replies. So, I have another question. Our original plan was to take the train to Derry and Portrush, then rent a car to tour around, then take the train back to Belfast. So I am just wondering if it would be better to pick up the car in Belfast and just drive to Portrush, then Derry and take a different route abc to Belfast. We have a few days, so time isn't an issue. Just wondering if there is enough to see on the drive that would outweigh the train trip. Any suggestions?
The full scenic Causeway Coastal Route goes from Belfast to Derry. There is a lot to see in this area as it covers the Glens of Antrim and the Causeway area. The most scenic section is from Larne to Portrush including Tor Head scenic drive (great on a clear day) but other parts are well worth seeing e.g. Mussenden temple/Downhill is an iconic landmark as is Carrickfergus castle.