I need to brush up on Rice because I just don't get why they are attractive. Without looking it up, my perception is that Rice is a strong academic school with mid-major sports and football traditionally a notch below Marshall and Southern Miss. No idea if that is accurate but there aren't money schools who would have a better chance of getting my wife to say, "Who's that? Am I supposed to be excited about this?" as we head to a game.
Rice was a huge power in the old SWC back during the 1940's and 1950's. They won a few conference championships and went to the Cotton Bowl and other major bowls. The City of Houston built Rice Stadium for the school after they won the SWC in 1949. Incredibly the 75,000 seat stadium was finished in less than a year for 7 million. They played their home games there the following season. UH played all its home games there until the Astrodome opened in 1965. By the 1960's they couldn't compete anymore in the SWC due to their high academic standards. Over the years they have produced some fairly good NFL talent.
Starting in the late 1980's they enjoyed something of a revival in football. For several seasons at the tail end of the SWC they were competitive. They even beat UT for the first time in forever during the 1994 season. They were in the WAC for a while and then joined C-USA. They've been up and down, but they have generally been competitive in C-USA more often than not. They broke a 40+ year bowl drought in 2005, and have been to one or two other bowl games since. Not that it matters, but they have one of the best baseball programs in the country. They won the CWS in 2002. Basketball has been awful, and they play in a glorified HS gym built in the early 1950's which used to be known as "The Jungle" because it wasn't air conditioned up until about 10 years ago.
Their biggest problem is the school only has an enrollment of about 3,000 undergrads, so they don't have a huge alumni base like their regional counterparts, but what alums they have are generally well heeled. They've also been lucky if they can draw 5,000 for home football games. They deliver next to nothing in their own media market.
Unfortunately, they would add nothing to the AAC other than their nationally prominent baseball program, and Wayne Graham is 80, so he's not going to last there much longer.