Note: Many people have come up with ideas similar to this, this is simply my take on how an item like rope would behave in Minecraft.
I think rope would be a really innovative and useful tool. You could use it by right clicking on two different blocks that aren't touching any other blocks horizontally, like a tree, or a pillar (this is because you need to tie the rope around the block). The first right click ties one end of the rope to the block that your cross-hair is on. Now you can walk around with the other end of the rope attached to you. The second right click will attach the other end of the rope that is connected to you on to another block. Between clicks you could wrap the rope around other blocks, if you wanted. Also, perhaps rope could be stacked, and the bigger the stack, the longer the rope could get (how much rope you have left would be indicated through the number of rope in the stack). To pick one end of the rope back up (detach/untie it) just simply hit it (left-click), but it can only be untied with your fist or another rope. Hitting it with any other item will break the rope on first hit. Destroying a block the rope is wrapped around will break the rope. Placing a block in space rope is occupying will break the rope. Players and mobs would be able to easily walk through rope without breaking it.
An upgrade of the rope could be the chain. It would pretty much act just like the rope, except it would be fire-proof and maybe less vulnerable to explosions. This could also be crafted into chainmail.
Tripwire (attach each side to the closest block to the floor, and enemies may trip when walking through it)
Zip-line (maybe you'd have to make a coat-hanger, but it'd be pretty cool transportation, right?)
Scenery (who knows what cool aesthetics people could make with this)
Mechanics (elevators and stuff...this would require new blocks like pulleys, however)
Leash (tie to an animal/enemy to easily keep it in an area)
Bread-Crumb Trail (easily find your way out of an endless, winding cave)
Fuse (enjoy the fireworks from a distance)
Mine-Cart Trains (pretty self explanatory)
Boat Thing? (you could probably do something with boats, right?)
Rope-climb (connect to a block only touched by one or two other blocks and touching no blocks vertically, and jump underneath to go into "rapple mode", where pressing up and down will let you climb slowly; upon reaching the end of the rope you will fall; not sure of any other way to get off)
Grappling-Hook (crafted from rope and arrow, once shot from bow, one end of rope will attach to any surface)
Rope-Ladder (crafted from rope and stick, once placed, will automatically extend down until it hits another block, or until the stack is used up)
I believe rope wouldn't be that hard to program. I don't know the details, but I'm pretty sure that making a line between two blocks shouldn't be too difficult. Attaching rope to movable objects may be another story, however. In any case, no physics would be involved. I imagine it would be similar to the fishing rod's line, that is to say, it wouldn't become a 'block' once placed. The only graphics required would be a single image that constantly faces the player around an axis, like flowers or mushrooms. Perhaps another graphic would be required to show which block the rope is tied to. Wrapping the rope around blocks would involve simply creating an angle where the rope comes into contact with the corner of a block.
Here's a possible crafting grid for rope: [string] [cloth] 
[cloth] [string]  = 4 Rope?
[string] [cloth] 
For chain, it'd probably be similar but involve
Any other ideas for uses of rope? Let's hear it!
edit: come back 2 years later...I still think this should be in Minecraft
Yet another thing I thought of but forgot to post. Rope would be just great! Not only could you craft with it, but the contraption possibilities are endless! I've learned how much a bit or rope can do from Gmod.
Ahaha... Leave Rope "bread crumb trail" accidently walk over lava... Rope burns leaving you lost in a limitless cave.
Rope would be good, would make Ladders more fun to make, instead os just being like "logs, wood, wood, sticks, sticks, sticks, sitcks... Ladder ladder ladder..."
Thanks for your input, Forces. Now that i think about it, making rope out of two different materials (string & cloth) could make it easier to craft, since getting string can be a pain (damn spiders!).
I also agree that there are almost infinite possibilities with rope, and I don't think that rope would damage the game-play at all.
I'm not sure what you mean by states...the rope could just be 1/4 X 1/4 blocks thick (or 1/8), or it could even be a 2D image that constantly faces your view, like flowers.
Again, I'm no programmer so I'm not sure how hard it could really be to put this idea in...
I think that this would be an interesting thing to have. you could use it to link mine carts ( I would say that the rope gets used up and the mine carts are just right next to one another but I could finally have the train engine in the front so I could feed it coal.
Also as far as climbing up and down goes, I think latters have us covered there. unless you want to go with pitons as well and the latter could even be put on the ceiling that would be different.