How to get from O'Hare to Midway (with family in tow)?

The City of Chicago as a web page on Transportation Between O'Hare and Midway.

At that hour, a taxi or car service will be your most expeditious option. At $75 cab fare ($25 per person excluding the 3-year-old), it will be the most expensive, but there are taxi stands at each terminal, the ride is only 40 minutes in light traffic, and you won't have to worry about train tickets or where to transfer or how to drag your bags up three flights of stairs if an escalator is out of service.

I believe shuttles are required to pick you up from the Bus Shuttle Center, near the CTA station and similarly accessible from Terminals 1, 2, and 3 via the underground passageways. A pre-arranged van like Omega Shuttle would probably be less expensive than a taxi or limo, but there is always a risk with shared van services like Airport Express or SuperShuttle that you'll have to wait for another party that's been delayed, or endure some drop-offs on the way. The Tri-State / United Limo service only operates once an hour, but has a "kids free" promotion that might save you a $23 fare.

ORD-MDW is one of the easiest airport-airport transfers you can make with public transit; however, you do need to go all the way downtown to transfer, there are a lot of escalators, and a lot of opportunities for delay. During rush hour, traffic delays make the train trip competitive with a taxi, but not at 8pm on Saturday. To get from terminal to terminal will be a bare minimum of 90 minutes if you make all your connections perfectly and see no other delays. Remember, online trip planners only calculate ride times according to published schedules, and don't account for customary delays or construction.

1. The train doesn't stop in the terminal; you need to walk over to the O'Hare CTA station. It's not an especially long walk (except from Terminal 5— take the tram over), as you can take an underground tunnel which has moving walkways... assuming the walkways are working.
2. You'll need to purchase a Transit Card. You only need one transit card for the whole family (you still need to pay the $2.25 fare for each person, but you can do it from one card), but I have never not seen a line, so that is another few minutes of waiting.
3. Blue Line headways are about every five minutes on Saturday evenings, but it is still a 40-minute ride to the Loop, then you need to transfer at Clark up the escalators to the Orange Line (taking care not to board any of the other trains that stop there). Orange Line headways are 10 minutes. Then it is another half hour to Midway, then up to the pedestrian walkway to cross into the terminal.

None of that is fun with luggage, a car seat, and a toddler.

Generally speaking Mass Transit in the US weekend schedules are lighter then the Weekday ones so you should keep that in mind.

You can take a look at Google Transit as CTA's page suggests at how you can get from O'Hare to Midway, which is not 1:15 but closer to 2 hours and with a transfer from Blue to Orange Lines.

It also suggest options using buses but again time is close to 2 hours and at least 1 but more often then not 2 or more transfers.

Then there are shuttles like: Omega, or Coach USA that run once an hour and take about an hour to get from O'Hare to Midway.

Then of course there is always taxis with fares running from $60 - $75, so pick your poison with your likely cranky and tired 12 year old and even more cranky and tired 3 year old.