Michael Watson gave what I think is the simplest answer in his comment.
.twitter-typeahead { width: 100%; }
Update #1: Based on the comments below (thanks Steve, Vishi), do the same for .tt-hint, .tt-input, .tt-dropdown-menu.
Update #2: mlissner reports that .tt-dropdown-menu is now .tt-menu, so the end result is
.twitter-typeahead, .tt-hint, .tt-input, .tt-menu { width: 100%; }