Steve,
No need to avoid Fremont Street. I've stayed at the Vegas Club, Four Queens and Binion's, and wouldn't hesitate to stay at the Plaza (on Main at Fremont), the Fremont, Golden Nugget, Fitzgeralds, Golden Gate or El Cortez. Of all of those, the Golden Gate is the oldest, and the smallest. The El Cortez is a block away from the Fremont Street Experience (a canopy that covers Fremont Street that is a massive sound and light show). If I were to book a room at the El Cortez, it'd be a tower room, not a vintage room nor a pavilion room. Just off Fremont, a block west on Main Street are the California and the Main Street Station, both suitable casino/hotels. A block east on Main Street is the Nevada Club. Avoid that one, it's a hangout for drug dealers and prostitutes.
...and I don't think you'd be disappointed with Arizona Charlie's Boulder. The rooms are larger than the average hotel room, and as I said, good inexpensive food in their restaurants. They have a sister property, Arizona Charlie's Decatur, but a first-time Vegas visitor might want a GPS system in their car for that one. It's a nice locals casino/hotel, but not really close to anything.
Two other possibilities that haven't been mentioned are the Stratosphere and the Sahara on the north end of the strip. I've stayed at both, and they both frequently have reasonable room rates. There's also the Circus Circus, but that is more of a family-oriented destination -- lots, and I mean lots, of kids running around all of the time.