Coleus usually will root, but sometimes they just rot like that - no idea why but it might be just something as simple as that stem was nicked, or not healthy in the first place, or some bacteria or disease organism was on or in the stem.
Your daughter's plant is really gorgeous. It will make lots of different colors as the weather changes. Don't go pinching off leaves just because they aren't perfect, though. Little things like moving the plant from bright light to dim, or change in temperature (had some cold nights yet?) can make them grow a little wonky but they're still feeding the plant.
Plants, like people, are never "perfect" and sometimes they do just die, with no apparent reason so keep that in mind as you grow your plants and learn.
