Stay away from most of the tower hotels west of the 405 freeway on Century Blvd. as most are pricier almost on-airport properties and at a few of them that I've stayed at departing jets rattle the windows all night.
Of all of them, the farthest away from the runways and likely the cheapest is the Travelodge - I haven't stayed there but drove by it often. There's a Denny's in their parking lot and it's always open - I've eaten there.
${hotelName} | ${city} Hotels, CA 90045
It's not low budget but there's a nice Embassy Suites just south of the airport that's pretty quiet - and offers free full breakfast. And a shuttle. Get a room on the south side, it faces a neighborhood. About $140?
It was my hotel of choice at LAX - I spent a a few days a month there on business regularly. They added a freeway interchange across the street a few years ago and while it cuts down the view, it also lessens the noise since the cargo terminals are on that end of the airport and pretty much go all night.
Motel 6 just on the other side of the 405 on Century is probably one of the better deals - it's $80 and they have an airport shuttle - don't know much about it except what I've seen from the street.
If there's a crappier part of the greater LAX area it's just north of the airport - some of the smaller car rentals are there and cheaper older hotels. Might be OK for one night since you won't have a car to deal with - parking is extra at many LAX hotels and you're not likely to leave the property - it's not an area I'd want to walk around at night. I don't know any of them or if they offer breakfast.
fwiw, my info is older, I haven't been thru LAX for some years. I doubt it's changed much except gotten more congested.
