In the new sites drop-down, there is a text input field that filters the list when typed into. It's called "find a Stack Exchange site".
However, to use it, you have to actively click inside it.
There is nothing else in that dropdown that accepts input. At least in Firefox 25, the find on page function doesn't care whether the text input field is focused or not.
So the extra click seems unnecessary. Keyboard scrolling is broken in more than one place on Stack Exchange anyway... (pressing Home when on a question page seems particularly unpredictable).
Hence I propose that the find text input field in the StackExchange dropdown is focused automatically when the dropdown is opened.
This is not a duplicate of Focus search field by default (check the dates