Dripping supply line from the toilet cut off valve

Knowing where the water lands gets you in the area, but what you really need to know is where the water came from. Is the tank cracked, is the supply line failing, is there a bad connection on either end of the supply line, or is the valve itself going.

If the tank is cracked, you'll have to buy a new one, and it's a large project if you're not used to working on basic plumbing jobs or are skirmish around an open toilet.

A bad supply line can be easily replaced and it's even easier if it's just the washers.

A bad valve may just have some dirt or buildup in it that will be fixed by opening and closing it once or twice with water running through it (an empty toilet tank). Though keep in mind any dirt may then get in the toilet valve and could cause it to run slowly until you clean it out. If that were to happen, just post another question. It could also be the packing material in the valve, which I've never touched on this type of valve. If that's the case, you may be able to minimize it by opening the valve all the way. If the valve is too far gone, you'll likely need a blow torch and solder to replace it. This can be a DIY job, but if you get in over your head, the water for the whole house will need to be shutoff until you can get a plumber out.

It seems as if you have an obstruction that may be floating around in the pipe or valve. Sometimes it obstructs sometimes it flows clear. I would start with changing the water valve. If this is the old style "gate" valve there are several things that can clog the line. I would replace it with a 1/4 turn ball valve. That will give you much better flow than a gate valve. You will need to shut down the water probably at the main. The old valve is probably soldered, but you could cut it off and use a compression type valve instead. "Sharkbite" is one such supplier. Although I have never used their compression valve, I have used their end caps and found them to work well.

If you can have someone hold a bucket under the pipe once you cut off the valve and then a second person turns on the water main to fill the bucket to purge the line of any debris. With cast iron pipes it is common to get clogs from rust build up. The rust may not pass the small opening in the old gate valve when you first purged the line.. Although you did not mention what type of pipes you have, if it's copper than you won't normally find rust obstructions..