Hans Wegner woven cane - restoring dry cane  

Lucky you!
I think it would be safe to mist or sponge the back of the cane with plain water to get some moisture back into it thought i'm not sure how much this will reverse the dryness. It may just keep it from getting dryer? I don't know.
Not sure oiling is the way to go but maybe someone here knows more than i do.
You can weave new strands in to replace the old ones. Just soak the new strips until they are pliable (15-20 mins max) and weave them into place, tucking the ends on top of the old, broken ends so that the new end is hidden under perpendicular strands. Do the same with the other end or wrap it around the rail and tuck it in wherever is appropriate.
New cane will not match the old, aged cane, obviously, but you can mess around with tinting the new cane. Just make sure you tint it with some kind of permanent color and let it dry completely before soaking. I don't think you'll be able to do much with the color once it's damp. Coloring it after it's woven in will also be tough.

Spanky is right. How lucky you are to have acquired such lovely family pieces!
Unfortunately, though, there is no effective treatment to revive embrittled cane. Once the fibers have begun to separate, it's pretty much done. The usual course is to re-cane and allow to age naturally.
Also, general consensus is to avoid treating new cane with anything to try to prolong service. Sometimes restorers will tone new cane by applying either shellac or lacquer with added stain or dye after it is woven to more closely match existing aged pieces.
Perhaps, if you've got good, basic hand skills, it might be a good opportunity to learn a new one? Cane and a few tools are relatively inexpensive and, if it doesn't go well, it's easy to undo without harm. Lots of instruction and good advice available on the web and right here!

Nice pieces. I love looking at my early Round chair every day, but unfortunately never sit in it due to the same brittle cane condition that you have. Since most of the cane looks pretty good on your chairs, it's tough to make a recommendation to replace. It would almost have been better to get a chair where the cane is trashed beyond repair, so the decision would be easy.

Except that this way the person redoing the chair has excellent information about how the chair was caned to go by to redo it properly.
Not too long ago I saw a Round Chair that had been taken to a caner, and the owner asked for the cane to be done as durably as possible, with no regard for the original pattern. The original pattern certainly was not respected, but I am not sure the result is any more durable. And the insides of the rails are now littered with staples.

Here is a video from Fritz Hansen where they do a partial repair on a Kjaeholm PK22. Although this is done with wicker-a similar technique could be used. Granted, the match between the new and old cane will not be spot on, but it might make for more usable furniture if that is your goal. Also if you are looking for good tips on re-caning, check out this thread
http://www.designaddict.com/forum/General-discussion/Replacing-Rattan-Bi...