What are Jigglypuff's strengths?
Jigglypuff is a quite unconventional character, that's true - but whether a character is good or not is very subjective, so it's very hard to give a proper answer to this. I'll try my best, though. When one has gotten used to playing as Jigglypuff, she can be quite a formidable fighter.
Almost all of Jigglypuff's B moves are very tactical, and take a while to learn. They do, however, work very well in conjunction.
B + UP: Sing. Sing takes a good while to execute, but when in close quarters without much danger of projectile attacks, this is a great move to use as it will put opponents to sleep. While they are sleeping, you often have plenty of time to inflict damage or try to smash them away.
B: Rollout. Rollout is a great move to start charging while the action is going on elsewhere, since a successful rollout hit can do a whole lot. Without charging, it can be used for horizontal movement, but isn't much of an attack.
B + SIDE: Pound. Pound is, to be completely honest, not very useful, though it can be used to move fast and far horizontally in the air, if used repeatedly.
B + DOWN: Rest. Rest is, without a doubt, Jigglypuff's most powerful move. It will put Jigglypuff to sleep for a good while, so make sure you're in a good situation when using it. If Jigglypuff is right inside (or at least touching) an opponent while initiating rest, it will smash the foe with incredible force. Just make sure your target has a high damage percentage when using it, and that not too many others are close by! If used together with Sing, it can often be easier to hit with.
Together with these, you can always use the quick direction + A combo (or the c-stick on a GameCube controller) to inflict some nice damage and possibly knock your foes a good bit away, and of course don't be afraid of going for a few grabs every now and then - Jigglypuff's grab + down can often work out quite well.
Jigglypuff's final smash takes some getting used to - it is probably most effective if your opponents are close to an edge.
If you think Jigglypuff is slow, you're not playing her right. You need to stay off the ground as much as possible to utilize her second-best airspeed, not her last-place walking and running speeds. While in the air you can then chase, bait, and weave around opponents using five double jumps and reasonably fast aerials. Your KOs should mostly be edge-guarding and gimps rather than smash attacks and the like.
Don't overfocus on special moves like the other answer currently here does. Pound is a critical part of Jigglypuff's recovery and can be used to bust shields or tag the ends of dodges, but Rollout and Sing are highly predictable and not very useful in a 1v1 fight (often, at even medium damages, opponents you put to sleep with Sing can get up and smack you before the move is even done). Rest can't be used for surprise low-damage KOs like it can in Melee, especially against characters with high gravity and/or falling speeds, and generally is not as rewarding to land anymore. If you're playing with items, you can sometimes guarantee a KO by using Puff Up on the edge when the opponent's offstage or close to it, but this isn't trivial.