The schedule change suggestion is great advice. Although the connection will probably remain legal with a small schedule change, you can likely find an understanding agent who would move you to the previous UA flight from MSP at no charge.

As for standby -- that's tricky. For one thing, it may be difficult to find 5 seats on the same flight to stand by -- although if you can get even 3, having one parent take the two youngest might be an alternative. However, unless you have UA Gold status or above, there's a $75 fee -- per ticket -- for standby if it clears.

OP: 40 minutes to 52 minutes doesn't move the needle much for me, but it does take it from "heck no" to "if I really have to, I'll chance it." You know your kids best; how quickly are you going to be able to move them across an airport? Are they going to need to use the restroom at ORD? I'd bring snacks from home, and buy whatever liquids you need at MSP -- not only will you avoid the hideous ORD markup, you'll also save time.

UA operates out of two terminals at ORD, spread across four concourses. For the most part, larger planes will operate out of Terminal 1 -- the B and C concourses -- and regional jets will operate out of Terminal 2 -- the E and F concourses. The Lufthansa flight will depart out of Terminal 1. B and C are connected via an underground tunnel with a moving walkway. What type of aircraft is your first flight? If it's a Boeing or Airbus (3x3 seating), you'll likely have a much shorter walk than if it's an Embraer or Canadair plane (1x2 or 2x2 seating).