Mid-July is when I took my oldest dd for her bday a couple years ago. Be prepared for torrential downpours each afternoon. It was pretty hot anyway, so we never minded getting really wet. And one day, we were hiding under the overhang of a kiosk with a group of foreigners and souvenir pens (the big, thick ones) started floating by from somewhere. My dd had a blast, splashing around in the "wading pool" the rain created, and we got free pens. No employees could figure out where they had floated from (probably had been dropped, and left underneath sales counter somewhere) so we got to keep them. We gave one to each family that was nearby, though, and kept one for ourselves.
Also, if you are stying on resort, we took the disney transport back to the hotel for lunch each day (by 1pm or so). Most families (with crabby little kids, mostly, who needed their naps , lol) were coming back around 3pm while we were done swimming at the hotel, eating lunch, and heading back to the parks. Worked well -- we got a seat on the bus, and avoided the afternoon return-to-resort lines.
If you want, I can dig up a few pix of the day the pens floated by, just lmk!
Beth