Avoid the supercharged V-6's (Ultras, SSei's, a few of the Olds 98/88 series) as they burn premium instead of regular.
Buicks and Olds are "old people" cars but that makes them cheap and also if you get a "grandma car" that's been well-maintained, you make out on that end as well.
Olds Delta 88's will go for the cheapest of all since Olds got discontinued as a brand name.
Going upscale a bit with Park Avenues costs a lot more. Avoid them as they're the same car as the LeSabre other than a few goodies being standard like dual-zone A/C, automatic temp Climate Control and such. 98's can be had fairly reasonably since they're Olds. Bonnevilles come in all sorts of trim levels. For dirt cheap and also a chance to get away from power window costs, find the low trim basic Bonneville instead of the high-line SSe.
Watch the Craigslist for your area as well as the nickle ad sheets. Most people selling them cheap are going for the free ads instead of the standard newspapers or AutoTrader to sell their car.
One time I read the least-stolen car was a Buick LeSabre...LOL! If no one steals it, the cops don't hassle it, if it's economical, comfortable and reliable, what better way to go cheap!
Rick