I have run The Wizard's Amulet (a 1st level adventure with some nice pregen characters) from Frog God Games and The Mystery of Malvern Manor (1 2nd or 3rd level adventure). Both went very well.
Since somebody already mentioned Malvern Manor, I'll note that I wrote a sequel as well. The much less appreciated Journey Through Crocodile Crossroads was also a hit with my kids, though we didn't get quite as far into it before they all had to go to bed. Or, if you're just looking for a story starter, you can try The Demon's True Name.
For what it's worth, I also ran The Lost Mine of Phandelver for my kids, and we had a great time with it. In general, I prefer to make my stuff up as I go along, but there are a couple of quick, fun dungeon crawls in Phandelver that are really terrific.