I just went through this, and it was very difficult.
I figured out from trial and error that the schedule loads into the system 330/331 days in advance, and if I called starting around 10:15pm PT, I might be able to get it right when the schedule loads (so far me, it was 331 days in advance, by an hour and a half, but 330 days per Central Time). Sometimes it didn't load and at, and I even waited up until 1am in the morning waiting for it to load -- I'd call every 10 minutes or so to see if it was loaded, but it never did.
I think AA.com is 2 days later, and therefore it will be impossible to get a Friday/Saturday night return if you are counting on AA.com.
What I did is that I booked my June 12th outbound, by calling at 10:15pm, and continuing to call until the schedule loaded, which that night was around 11pm. They held it for 5 days for me, but would not extend beyond that date for me when I called a few days later to try to extend.
Accordingly, I issued the ticket (paid the fee for talking to a person) with a June 12th outbound and a return on June 17th (which I never intended to take).
I then waited a week, and started to call again, initially getting a return on June 23rd, then calling the next night and pushing it to June 24th, and then finally calling to push it to June 25th. I couldn't get June 26th, my desired return date, because someone else was doing the same thing, and grabbed the seats right before I got them. Keep in mind -- this was 10-15 minutes after the schedule loaded, and the 35,000 FF award tix were already gone.
What was interesting/frustrating was the inconsistency in when the schedule loaded.
I will also point out there appeared to be better availability on the OGG/LAX redeye flight versus the afternoon, as well as the HNL/LAX flights, both afternoon and redeyes. For me, it was just the afternoon flight OGG/LAX that appeared highly restricted, but I was trying to get it for my young children. OGG/DFW always appeared accessible.
Good luck, and don't get discouraged. AA representatives (more than one) did tell me they expect to get more tix FF loaded over time as they better understand the loads.