A neighborhood kid caught a baby alligator snapping turtle somehow (no lakes, ponds, or creeks on our area). I stole him since he wasn't going to take good care of him. My brother's friend is going to take him to a large lake near his house, but he hasn't come yet.

The problem is, the turtle hasn't eaten for a few days. His shell is also very soft. He swims plenty- I have a few weed leaves in there in case he'll eat that, rocks for him to get on for when he wants out of the water, and a large aquarium. I tried to feed him a worm, but he wouldn't eat it.

I have a few minnows in my pond. Even though I hate to catch one to put it to it's death, but I guess I have to if I need to nurse it back to health.

I don't want to release it in such poor condition. What can I do to get his shell re-hardened?

PS- I'm pretty sure he's an alligator snapping turtle. He has spikes on his shell, but I can't tell of the pattern since he's dark. He is definitely a snapping turtle, though.
One more thing- I picked him up and under his shell there was some brown stuff coming out. Was that poop? (God, I sound like an idiot.)

Hatchling turtles are born with slightly soft shells- it is normal. It's not the same as "soft shell" per se, which afflicts captive turtles that are not properly cared for.
You probably have a common snapping turtle- they are much darker than gator snappers, and more easily found. A google image search would help you identify it.
Re the brown stuff- if it came from the cloaca (anus) it was poop. If it was on the center of the plastron (belly button), then it was the remnants of the egg yolk, which is absorbed within a few days after hatching.
The best thing for the turtle would be to release it where it was found. They are a difficult for beginners, and if it survives until adulthood, it will be difficult to handle, and require a very large enclosure.
Please do the responsible thing.

If you have a hatchling, don't worry about the soft shell, If the turtle weighs in at 100 lb, that's another matter.

When young, even the common snappers have spiky shells. Look at the carapace (upper shell). There are three rows of large scutes and a ring of smaller ones around the edge. Look at the side, between the larger and the smaller ones. Do you see three extra scutes? If not, you have a common snapper. If so, an alligator snapper. Location might help to tell them apart too.

Either one should eat fishes; the common snapper would be more inclined to take the worm. It might take awile before it feels like eating.