The problem with building a web browser is that if you introduce too many new features then you are breaking with web standards. The most you can do is add some user conveniences like sync, themes and extensions, but those don't go far enough to make enough users consider switching. A browser can, by definition, never have a "killer app".
There is so much more a browser can add beyond just rendering websites "correctly" - and even that does not mean always rendering them the way the designer wants. Just a few things that could win over power users:
- Better tab organization. Vertical, tree-style tabs are an improvement possible with extensions, but surely we can do even better, especially if with direct support from the browser.
- Better bookmarking. All bookmarks should take a snapshot of the website in case it disappears or changes and should have fast full-text search. And those are only the obvious parts that are missing.
- Integrated ad-blocking and user scripts/styles to enhance and fix user-hostile websites. Make it easy to share these with non-technical users. Really anything that puts the user in control. This includes the Browser UI too - firefox has been going backwards con customizability.
- Ability to keep up with websites directly through the browser. Aka bring back RSS support and actually make it useful.
- Anything that works against the centralization of the web. Allow users to comment and share websites with their friends or community right from the browser in a way that is resistant to censorship.
And this is just what I could think of right away.
