My preference changes with my mood and what I have found on the Seattle or Columbus (now leaning toward starting there and doing two complete loops) craigslist lately, but there are a few cars that I keep coming back to. They fall into two categories...
Roadsters:
Miata - A mechanically solid, slightly rough looking NA would probably be the best fun-for-the-buck answer on the sporty end of the scale (at least without running some serious reliability risks.) An MSM or early NC are stretching my budget, but would be nicer on the highway and still all kinds of fun on the Beartooth Highway (as long as it wasn't three feet deep in the snow.)
MR2 - I have owned an AW11 before and it has been both my favorite car to drive and the best value I have ever gotten out of money spent on a vehicle. Not as forgiving to drive in bad weather though. There is an extremely clean SC version in Seattle, but it is optimistically over-priced at $9.5k. I've also owned an (NA) SW20, which I didn't enjoy nearly as much. It was boring unless thrashed within an inch of its life. I've test-driven a couple of Spyders and have been highly tempted (would have bought one if I hadn't found the '86 for $1500.)
S2000, Z3/Z4, or Boxster are all appealing, but perhaps a little rich for my blood in various ways (initial coast, potential maintenance, higher strung, getting maybe a little too much attention on the highway or parked at colleges and cheap hotels.)
GT-ish:
E36 M3 - I'm shocked (shocked I tell you!) that we're into page three of this thread and this hasn't been mentioned except in the fecal sense. It has some of the same downsides as the higher-end roadsters, but at the very least I intend to take the excuse to test drive one or two.
Legacy GT - I like Subies. AWD is good in the mountains. Sporty, fast, Q-shippy wagons are fun. People also don't seem to want to sell them. (I suppose I could live with a sedan.)
V70R/850R/T5 - Sporty, Q-shippy, fast wagons... still fun. Surprisingly easier to find than Legacy GT wagons.
Late C4 or early C5 Corvette - This may be one of the smartest, most economical ways to cruise the highway in fast, fun comfort. I respect the hell out of this car, especially the newer generations, but... there is a part of me that expects Vettes to be driven by arrogant middle-aged shiny happy people and does not want to be that guy.
Others that I have considered in roughly descending order of interest:
Other BMW E30-E46 3-series - Plenty of them around in a variety of conditions, occasionally even in manual. Would be nice if I knew which all of the cornucopia of models were actually significantly more fun than a Camry.
Subaru RS 2.5/WRX/Forester XT - Ideally a wagon, not thrashed or haphazardly modded.
5-series/S-Class wagon - I have a general sense that this class of car represents more upkeep cost than I want to take on, but I might accept that for a powerful, fun wagon (I would rationalize that further, but I know GRM gets that.)
Mercedes 300D... or older SL - These are terrible (for lack of driving excitement on the one hand and risk of high costs on the other,) but fun ideas.
P71 (other panther chassis car, caprice, etc...) - Spend next to nothing, easy cruising power, never get pulled over, take a moderate hit in gas and still come out ahead in most ways. Definitely not a very sexy option, but the upsides aren't bad.
Pontiac Solstice GXP - Might need significant attention if it is in my price range. But I'd go for any of the above roadsters before getting the non-GXP version.
350Z/G35/RX-8/Genesis - When the 350Z and RX8 were first released they would have been at the top of my list. Not sure why they don't appeal so much any more, other than they seem like a compromise between the M3 and the Miata. I'd rather have one end of the scale or the other.
Audi TT - If I fit comfortably (6'1"), might go on the roadster list.
VW Jetta/Golf/Passat - GTI, TDI...? various mixes of efficient and fun. Not sure which ones in particular I should be looking for.
Saab 9-3
Mazda S/Speed sedans & wagons
Acura RSX/TL/TSX
Jeep (Grand) Cherokee, Trooper, CRV, etc... - I can certainly have my pick of Jeeps in Ohio, but again, no idea what I would be looking for.
Del Sol
Corolla S/Civic SI
Volvo 240 GL - Poor man's 300D? (Mercedes fans seem to hate the comparison, but it certainly makes sense to me.)
Accord/Altima
Insight/Prius - MPG, MPG, MPG, but not sure I see any reason to do this in a world with TDIs
Mustang - Like the Vette, my opinion of this car is much lower than it deserves. I'm sure there are Mustangs that would be fantastic for this trip, but I have a hard time disassociating them from a certain class of Midwest poseurs from high school and college.
Well, aside from having a hard time finding Legacy GT wagons (except for a B Spec... far out of my price range) I certainly can't complain about a lack of options!