Hence my name, I'm a total newbie, so please forgive my question which probabl sounds stupid to you all.

Quick background information.. My father-in-law bought a small bowl and two standard goldfish for my 3 year old exactly one year ago. I've changed the water weekly with distilled water, made sure the fish were fed, and had good light for 10 hours a day. A month ago, I noticed one of the fish swimming more and more with his tail to the top of the bowl. Eventually the other fish did the same thing. Next I noticed that they couldn't swim to their food. Their side fins were almost stumps, and their tail fins appeared to be worn down.

Tonight one of the two fish passed, and I gave it a nice burial in the yard. It was just a $0.50 cent basic orange goldfish, but the little guy was pretty cool, and lived for a year. The other fish isn't looking too good.

I've read about fin rot here, and know that the other fish doesn't have much time left.

My question: Do I put the other fish out of its misery and euthanize him, or do I leave him in his bowl and see how long he can live without food?

Hope this doesn't sound like a stupid question, but the fish is a living thing, and I want to do what is right.

Thanks for your comments..

It is possible to save the fish so you don't need to consider euthanising them.
First of all, goldfish should never be kept in bowls. They just aren't big enough and the water quality is just impossible to control causing lots of health problems such as stunted growth, finrot, internal infection etc as you are realising now.
The best thing you can do for these fish now is to buy them a proper tank which is big enough for their needs. it is generally recommended that each individual goldfish should have 10 gallons of water, but in your situation almost anything is going to be an improvement. You also need to get a decent filter. Set the tank up and let it sit for 24 hours before moving the fish into it (float them in a plastic bag so they can acclimatise). If you carry on keeping the fish in a bowl then it's outlook isn't very good I'm afraid, although it might be possible to save it this way. Please get a new tank if you possibly can!
If you do get a new tnak it will go through the cycling process and during this time levels of ammonia and nitrite become toxic so testing of the water and water changes daily are neccessary. If you get a new tank let me know and I'll try to guide you thought the process.
For now though, let's make him happier in his bowl. Why is you use distilled water for water changes? Tap water is generally ok with the use of a dechlorinator such as Aquasafe. Start daily water changes of about 50% in this case. This will be neccessary for the whole time he's in the bowl if you want a chance of saving him. ALso get hold of some aquarium salt and in a glass with a little tank water dissolve one tablespoon per gallon of water. Then add it to the bowl gradually over 24 hours (in about 3 stages). When you do water changes you will need to measure the amount of water removed and when refilling the tank add the appropriate amount of salt back to the new water. So if you remove half a gallon you will need to add half a tablespoon of salt to it.
Get yourself a bottle of melafix (widely available at lfs's) and treat according to the directions on the bottle. You treat for a week then do a 25% change. I'd treat for about 2 weeks with it maybe three. Remember again that you will need to add some back after every water change otherwsie you will be diluting it.
Don't expect immediate improvement on the fins. It will take them ages to grow back, but you should notice that they don't get any worse.
It also sounds like your fish are suffering from swimbladder problems. From now on, soak all food in a little tank water for a few minutes before feeding. This makes it easier to digest and stops air getting trapped in the fishes gut which can cause this problem. Also, to help for now, starve the fish for 3 days. Then for another 3 days, feed only shells, chopped, defrosted frozen peas. This should clear any air out of the fish and help a bit unless it's progressed too badly. Slowly reintroduce your usual <soaked> food but always feed peas twice a week from now on as they will prevent problems. You'll find your fish loves them too!
Hopefully if you follow this advice your fish should make it. Good luck and just ask if you have any more questions.