What about leather? You can buy it by the spool, just like ribbon.

Cover it in jute, perhaps? There is a method which is almost like finger crochet which used to be used to cover coathangers with yarn, if you are familiar with it that's what i have in mind. I'm not talking about the fancy crocheted covers for thick hangers; I'm talking about the single chain around a wire. I've looked but cannot find decent instructions of this online. This is similar to what I was thinking of:
http://reduxdesigns.blogspot.com/2007/07/tutorial-crocheted-wire-hangers.html

Do you want it to be done professionally, or do you just want to wrap something around the handle?

Honestly, if you don't care to preserve the brittle plastic that is falling off, I think it would look beautiful as just pewter. It looks like the plastic was put on the handle as an addition and that the finish underneath is unscathed. If you don't want to keep the plastic, just let it go and then you'll have a gorgeous pewter tea pot. If you're concerned with burning your hand, I suggest potholders.

I second gettysgirl's suggestion. Easy-peasy, and beautiful! Here are some instructions on whipping - wrapping the lacing so that the ends are enclosed (instructions are for rope but the method is the same):

http://www.ropeworks.biz/reader/whipping.pdf

What goes better with Pewter than leather?

I have several teapots from Japan that have a similar handle covering of bamboo....I wouldn't know how to go out reproducing it myself but I know there are shops in Japan that still make them so it's possible...

Because of the lead content of antique pewter, it is not recommended to use it for cooking or serving food. It can contain up to 15% lead which can leach out into food and cause illness.

It would look great on a shelf!

I was actually thinking thick hemp finger crocheted around the handle (similar to the first two ideas). Hemp mostly because it won't smell when warm or wet, it washes easier than leather and most yarns, and is fairly inexpensive. Good luck!

The old wrap is vinyl -- sort of like the vinyl-wrapped patio furniture from the same era. You often see pitchers and teapots with handles wrapped in cane (wicker?). Also nice.

You could DIY this if you're inclined and a bit handy. Look for info online: Vinyl wrap repair, patio furniture restoration supplies, etc. Also, look under 'caning repair' or similar. I think it is pretty much all the same techniques, regardless of the materials.

If you use this, I wouldn't suggest leather -- because of difficulty washing.

If you want to replicate what is there now, any craft store has vinyl lanyard cord in many colors for maybe a dollar a bunch. You might need to use a drop of super glue to start and end the wrap, if you can't figure our how to fold the ends under the wrapping as probably was originally done. And you'd have to decide if you want to match the old cord or change to a modern fun color. And you'd need to decide whether to add the woven in contrasting color or not. (I'm pretty crafty and it would be easy for me. If you aren't, maybe a more crafty friend would help out.)

I'd suggest using a Teapot Handle Mitt:

http://www.grandmarae.com/handlemitt.htm

How hot does the handle get? Do you just pour boiling water into the pot, or put it directly onto a heating element? If you heat the pot, I'd be leary of the vinyl suggestions--don't want it to melt!

I suggest sugru

'cause it's silly and you can mold it to have finger holds in the handle

http://sugru.com/

Just leave the plastic off and have the metal showing. That would be the nicest thing to do. Use a potholder when picking it up.

Cost: USD 0

Fifthpocket - the pewter teapot in question, if manufactured in the last century, almost certainly does not contain lead.

I would love to entertain more. I have this fantasy about effortless entertaining, my house filled with friends all the time. Some people seem to be able to do this. I cannot. I entertain very rarely, in fact. With my busy work schedule as a writer, my house is a mess most of the time. Having people over is just too much work and stress. It's a major production. When I do have a dinner party, I'm usually sorry about it the day before because I'm in a panic about it and I have to work like a dog to get everything done. I'm usually happy with the result, but it's just too much work. Wish it was different.

Wow, that lead content post scares me. There is an artful way to tie a linen cloth around the metal handle of a teapot. I've seen it done. I haven't yet found instructions for this, but I'll keep looking, as a I have a silver teapot. If you work out a way to do that, you'll protect your heirloom from further wear.

thanks for the ideas!

regarding the teapot, it's 20th century so as far as I can determine it's not containing dangerous levels of lead.

I do use it as a teapot so it's not sitting on a burner but the handle becomes scorching hot when it's full of water. I've inadvertently grabbed the top to remove teabags and it's been painful.

I kind of like the leather idea but after discovering that this brand is somewhat collectible I think I might just retire it as a decorative object that represents the interesting history of my grandparents living abroad.

Another vote for sugru! The stuff is indestructible, washable, soft, formable and won't burn you.

Unfortunately, it only comes in black, blue, green and orange.

How about the cane they use to weave cane seats on chairs? I'll bet a local upholsterer could point you towards a good chair caner and your repair would be very reasonably priced.

Heath does a teapot with a similar treatment on the handle. I bet if you asked them, they would have ideas for how to get yours fixed. http://www.heathceramics.com/go/heath/tableware/store/index.cfm?catID=46

Not long ago a friend had this little sterling silver tea pot handle repaired here.
http://www.acaneandwickerfixer.com/wicker_rattan_%26_wrappings.htm