We jusr returned on Sunday and utilized e-ride night on Friday @ MK. The park was open from 10 pm until 1 AM for guests who purchased the additional tickets. The cost was $12 per adult & $10 for children and you need to pay tax as well. The following are the rides that are scheduled to be open: Astro Orbiter, BTMRR, Buzz, Country Bear Jamboree, Alien Encounter, Haunted Mansion, Space, Splash, TTA.
The dates for July (regular season) were July 1, 8, 11, 15, 18, 22, 25 & 29 so you can see they do offer it outside of "off-season" times. The tickets are available at Guest Services desk in any of the Disney resorts, they are sold on a first-come, first-served basis and you have to show your resort ID & a multi-day ticket to purchase them. You can also buy the tickets at City Hall in MK on the day of the e-ride.
You exchange your ticket for an e-ride wristband after 4PM at City Hall, Tomorrowland Arcade or Splashdown Photo. Only park guests that have wristbands are allowed to enter the rides after the scheduled park closing. They do look at everyone's wrist when they enter the ride line--especially right at the park closing time.
We rode everything we wanted on the July 18th e-ride night--even though it was a torrential thunderstorm! We did Buzz 4-5 xs, Space 2xs, HM 2xs, BTMRR & Splash too, so it was well worth the extra money.
I know you'll be there in August and the dates for august are: August 1, 5, 8, 12, 15, 19, 22, so hopefully one will work into your schedule! I think your idea of taking it easier during the day with your little one makes a lot of sense. The nights in the parks are not nearly as crowded (unless Spectro & Fits are taking place) and the place seems to have a different kind of *magic* IMO.
Anyway-hope this helps!