The biggest question is: what is your budget? Typically short term furnished rentals start at $200 a night.
Try to stay in Downtown LA (simply google Downtown LA and you will see the borders of the neighborhood. Try to stay closer to the western section of it). Or stay within USC campus limits. If you wander beyond that, the neighborhood gets very sketchy very quickly. Here are some options: "Oakwood at Olympic and Olive", "Residence Inn at LA Live", "Global Luxury Suites at Figueroa", "Corporate Suites Downtown"
If these are out of your price range, then your best bet would be to look up some Airbnb rentals.
If Downtown/USC is not an option, I would suggest either Culver City, Burbank, or Glendale. All of these are nice, walkable, and safe neighborhoods. As well, there are train lines that take you directly to USC. Burbank will probably have the most short-term rental options, as a lot of people in the film industry need to stay there short-term for work. Take the Metrolink in Downtown Burbank to Union Station, and the Expo line to USC from there.
I hope this helps, and your son makes a speedy recovery!!