It runs but doesn't sew can be any number of things. Like mermaid stated, if the needled bar doesn't move it can just need a new belt.
If the needle does go down but it doesn't pick up the bobbin thread that can be anything from as simple as a bad bobbin or needle in wrong to just needing to be re-timed. I say take the vintage machine in and get an estimate for fixing and a complete tune up where the tech will oil and lube her up. A vintage machine like your mom has is a diamond in the rough. Definitely give this great Singer a second lease on life.