It's happening already - the contenders are JQuery, Prototype, YUI, Mootools, Dojo, GWT, haXe, etc., along with Apollo/Silverlight/Parakey if you allow desktop plugins. We just don't know who the winner will be yet.
Personally, my money is on JQuery (literally - I'm using it for my startup), but it really is too early to tell. Interestingly, nearly all the "serious" contenders are open-source: it's possible that the developer community learned their lesson from the Microsoft monopoly of the 1990s. It's very difficult to get any sort of programming tool adopted these days unless it's open-source.