There are a number of surface lots in the vicinity as well as other parking garages, some of which allow overnight parking. Most will run ~$20/day and it's simply a matter of hunting for a spot. Another option, if you're not planning to use the vehicle during the stay, may be to use MARTA's long-term parking and take MARTA to the Marquis ($2.50 fare one way). I know the North Springs, Sandy Springs, and Dunwoody Stations on the North Line (close to GA 400) have designated long-term parking lots, as well as Doraville and Brookhaven on the Northeast Line (close to I-85). Rates are $8/day for North Springs and Doraville; the rest are $5/day. Both lines make stops at the Peachtree Center Station, which is a 3-minute walk from the Marquis. You can walk to the hotel on the street, or if you don’t want to walk outside, you can exit the station into the Peachtree Center mall and follow the signs to the skybridge connection to the hotel.
Surface lot parking is usually more readily available on weekends, but it won't be free, though rates are usually lower than weekday rates. And most lots have no in/out privileges -- you pay every time you drive out of the lot, with no guarantee there'll be a spot there on your return. We live about 90 minutes from downtown Atlanta, which means our trips there are always day trips, so MARTA works great for us. But in all honesty, if I were planning a trip such as yours, I'd bite the bullet and pay the hotel's rate -- it's hard to put a price on security and peace of mind. Enjoy the stay!
I think that the hours parking is enforced varies by area in Atlanta. Where we frequent near our GaTech daughter's apartment, parking in enforced from 7 AM - 10 PM Monday - Saturday with a 4 hour limit on meter. Holidays are exempt. Some public parking garages in the area are around $15 a day. I would agree with Foxglove and bite the bullet and pay the hotel. We stay at the JW in Buckhead and pay the $24 a day.