Peter Hvidt Chairs  

What do you mean by terrible...
What do you mean by terrible and in what way would you like to replace the seat, do you mean re-cane?

well
The woven grass seat is just 50 years old and well-used. It looks dirty and some strands are looser than the rest. Is it possible to re-weave this with a different material?

After 50 years
the seat material might be said to have proved itself, no ? Yes, you can have them rewoven, with the material of your choice. Nice chairs.

Check out this thread
For some inspiration.
http://www.designaddict.com/design_addict/forums/index.cfm/fuseaction/th...

woven seats
Yes, you can have the seats rewoven. Or you can reweave them yourself if you're adventurous. It looks like they're woven of seagrass, which is pretty easy to work with though it does stretch a bit over time and isn't as durable as other materials. It's kind of greenish-gold when new and then ages to a golden brown that's very attractive.
The other material that would be appropriate is Danish paper cord. This is harder to work with than seagrass, so if you have them redone by a professional you should make sure that that person has done this particular kind of seat in paper cord. I've done them in both materials and aced them with seagrass and failed epicly with paper cord. Just saying.
There's also genuine rush, which is the leaves of bullrush plants dried and twisted together, but that's not really used on Danish chairs that I know of. And there's paper rush, which is mock rush made of twisted kraft paper. This looks similar to Danish paper cord but it isn't plied and is not as durable. I would not use it. If you go with the paper cord option by a professional, make sure he or she uses paper cord and not paper rush.
Frank's Rush and Cane Supply in CA is a good source of materials if you want to try it yourself. They have seagrass in a bunch of diameters and Danish paper cord in several sizes and types. I like the laced myself.
Finally, expect to pay at least $100 per chair to have these rewoven. I've heard of prices over $200 in some big metro areas.

papercord
I'd actually really be interested in reweaving this in papercord. I have seen this particular chair redone in that material and it looks really great. Plus durability is important if we are using the chairs a lot.
Spanky, in your opinion why is papercord much harder than seagrass to weave?

Papercord is much stiffer...
Papercord is much stiffer than seagrass. The success of this type of weave depends on getting a 90 degree turn in the cord every time you turn it. Even a tiny bit of deviation turns into a big gap at the end. I read somewhere that you can give it a quick soak in water---like 20 seconds or something like that---to make it easier to bend it. Wetting it does make it more flexible but the plies tend to untwist when it's wet. I dunno. I have wove the seat of my Wegner J16 rocker two or three times with paper cord and could not get it right. Someday I'll attempt it again.
The video below is of the guy who does a lot of (all of?) the seats for Fritz Hansen. You can kind of see the effort he puts into getting the cord tight and straight. Or maybe that's only obvious after you've tried it and didn't get it tight and straight...? He's good at it and make it look easier than it is.
http://www.youtube.com/watch?v=ImKkFwcopQg&feature=related

Here's another video about...
Here's another video about how to weave that style of seat in what he calls paper cord, but it's actually paper rush. The video is good for showing how to do the basic weave, but you can see also that the paper rush is not all that tough to work. You can see that it flattens out at the turns and around the rails. Paper cord maintains its round diameter when you bend it and wrap the rails with it, which is part of why it looks so nice. But that's also what makes it harder to work with.
http://www.youtube.com/watch?v=AHu7lc5680w

great videos
spanky, oh thanks so much for the videos especially the second one. I see what you mean about the 90 degree turn. It will definitely take a lot of practice to get it right, but I am very intrigued by the idea of learning it.
The good thing about these chairs is that there are just the posts on the four corners to deal with. Your Wegner chair is probably much more complicated because of the back. I really do love the paper cord look, much more polished.

Nah...
The slotted back of the Wegner rocker didn't make it harder. It was just those damn 90 degree angles! Good luck with yours. Please post about it, whatever you decide to do.