One of the other closed stations (Brompton Road) has recently been bought I believe for such purposes. No doubt the admission price will be the usual tourist rip-off rates charged by other London attractions.
Aldwych opens quite often and for some reason is very popular and provides useful income to London Transport Museum. Opening it full time, the novelty would wear off. (I also believe that LTM charge too much for the tours but as they always sell out, what do I know?)
Having had a look round a few times, I don't see why there is such a big fuss about Aldwych. There's not much there that can't be seen for free (if you have a travel ticket) at many other open stations on the network. Some tunnels with tracks in, some passageways, a broken lift and a ticket office.
If Brompton Road or York Way or Down Street etc.. were to be opened in their current state then that might be interesting.