When you contact the rental car companies, make sure you specify the Port of Houston and not Galveston. If you just tell them the One Way rental to the cruise terminal, they may think you mean Galveston. But you should be able to get a one way rental from the Airport to the port. Book your hotel if you have not already. They ones closest to the port fill up very fast. If you can't get anything in La Porte, I have stayed at and recommend the Holiday Inn Express on HWY 10.. Its about a 10 minute drive to the port but its a very nice hotel for the price. If you need the address, post back and let me know and I will get the phone number for you.
Carnival Cruise Lines:
Fantasy 90, Celebration 95, Victory 05, Conquest 05, Spirit 6/06, Freedom 6/08, Holiday 7/08, Victory 6/09, Valor 9/09, Dream 6/10, Splendor 8/10,Conquest 8/11,Magic 11/11,Triumph 4/12,Spirit 7/12,Freedom 11/12, Triumph 2/13 - The fire cruise, Magic 1/14, Freedom x 2 6/14
NCL Dream October 2006, RCCL Mariner OTS June 2007, Celebrity Silhouette 2013