How should you cite dead urls in a thesis?
You cite/reference stuff so later readers can check it. A dead URL is mostly useless, cite the archived one if need be (and explain why! perhaps the page went away for a reason that makes your whole point of refering to it moot...).

You mention "web application", presumably the application behind the page went away for good, in which case you don't have much to discuss anymore. Either find some detailed discussion elsewhere that covers what you are interested in, and cite that, or see if the application still exists (under another name elsewhere, a evolved version, ...) and see to cite that.