Are there hotels near the hospital that offer a discount if you have someone that's a patient there

Yes, Sonesta is very generous!

I do not have funds to pay more than 50 for the weekend

Yes. Ask when you make your reservation or when you check in at the hotel. Sonesta is nearby and offered a $89/night rate, which is much better than the AAA rate. Convenient and the suites are large, with a full kitchen.