I think Rand will have a new hand before the Last Battle. I say this for a couple of reasons. As someone pointed out back on the wotmania theory post, Nynaeve once equated healing stilling to healing a severed hand, in that both were considered impossible:

"Stilling is not a thing anyone would choose to study, you understand," Nynaeve continued. "It is generally accepted to be irreversible. What makes a woman able to channel cannot be replaced once it is removed, any more than a hand that has been cut off can be healed back into existence." At least, no one had been able to Heal stilling. There had been attempts.
- The Shadow Rising, pg. 118, Questions

As we all know, Nynaeve did heal stilling, and I don't believe in coincidence in WoT anymore.

Secondly, as was also pointed out back on the wotmania theory post, Rand has some similarities to the Irish myth of Nuada Silver Hand, who loses a hand and has a silver one made for him by the god of healing as a replacement.

Now, connecting these two points, I believe Elayne will make a ter'angreal hand for Rand. Nynaeve cannot make a new hand of flesh, but she will be able to connect an artificial hand to Rand's arm and make it fully functional. Herein lies the similarity between healing stilling/gentling and healing a severed limb: she cannot create something from nothing, but she can reconnect something that has been severed, whether it be the ability to channel or a limb. The only question is what the ter'angreal that Elayne makes will do. Too much has gone into Mat refusing to let anyone study his foxhead ter'angreal for it to go unstudied over the entire series, and Elayne in particular has an interest in it. I think the ter'angreal hand she makes for Rand will have the same properties of Mat's ter'angreal. This will then setup the perfect showdown between Rand and Moridin, who will be using the True Power, which is most likely unaffected by such a ter'angreal.