When using Firefox, I realised I am forever performing the sequence Ctrl-T followed by Ctrl-K to start a Google search in a new tab. However, I find the standard new tab page useful, so I didn't necessarily want to change my new tab page setting (you can change the page by tweaking browser.newtab.url in about:config).
This sounds like a job for an add-on.
And so I built one. The input is conveniently autofocused, so one can just Ctrl-T and start the search. I guess I'll eventually want to make things like the search engine configurable, and it's not particularly pretty, but for now, it's a great help!
Edit: the add-on is now up on addons.mozilla.org.