She's effectively judgment-proof. They can't take away her house and she makes too little for them to garnish her wages so it'll cost RIAA more to proceed with the suit than it does for them to say they made their point, collect a few hundred dollars and move on. I wouldn't be surprised if they eventually settle for a fraction of the $4,000 offer in exchange for a promise never to illegally download songs from the web again.
If I were in the RIAA's position, I'd allow them to settle for $1, an admission of guilt, and the promise you mentioned. The "victory" matters more to them than the damages, and they certainly don't need the additional bad PR of trying to bankrupt a single mother.
I would say a majority of the people who stupidly settle are judgement proof, because of inadequate or Homestead-protected assets. Which underlines why NO ONE should settle. At worst they should call the E.F.F. or other groups who are chomping at the bit to set legal precedent on this--if ANYONE would fight the damn bloodsuckers.