Using my superior google powers, I found a list of every hotel/bed and breakfast place in Fargo. If you go to this link, you can see the list for yourself (I had issues with the site remembering the preferences. Just choose Fargo or West Fargo for the city)
Yeah. Fargo has like... no mom and pop places. ONE bed and breakfast (if its nice, its in the heart of downtown which is sweet but you get to deal with LOTS of college drunks if you visit after mid-august).
But not all is lost. I also searched above link for the much overlooked West Fargo. The Sunset Motel would be worth calling (they don't have a website). I don't know about extended rates, but I had family stay there for periods at a time and I know that its cheap, privately owned, the rooms are clean and they have a waterslide/indoor pool. The link above says they have rooms with kitchens also. It is currently on the edge of Road Construction Hell but that hopefully with be over with by fall.
The cheapest place in town (not listed on the site) is the HI-10 Motel in West Fargo. Don't stay there. Although obscenely cheap and they do extended stays, the place is run down and filthy and many sex offenders live there. I used to work at a restaurant that did deliveries and one of the guys told horror stories about the place and about a customer that (at the time) had been ordering to there for 7 years and lived with (his words) a dozen cats. It also is (was? maybe it stopped) the last place in town that still does hourly rates. You get the idea.
On the minnesota side of the river, Moorhead has a handful of chain hotels (the same chains are in fargo though) but the only mom and pop place I could find at all is one of those 50's L-shaped motels called the Star-Lite Motel in Dilworth. Wikitravel says its clean and inexpensive, but Dilworth is on the far end of Moorhead (and you probably don't want to drive through two towns to get to Fargo if you're a tourist).
Now that I've searched my little heart out, I demand more mom and pop places in fargo!!! I mean REAL mom and pop. Privately-owned franchise shares don't cut the mustard. :P
Edit: I found a few more places. There's a few B&Bs in fargo but they're pricey, even if they do feed you. Then there is the "Guest House Motel" in Moorhead. can find no info other than it does extended stays and the addy and phone number but a map shows that its right by a college campus (MSUM) and Romkey Park, a small park that sees far too much crime for its tiny size.