How can the Omnibar in Chrome be made to act like the AwesomeBar in Firefox? Specifically, Chrome doesn't seem to recall Google Docs from the history when I type some of the text from the doc's title (which isn't part of the url but only in the html title.)
I don't think a complete change of behavior is possible at the moment, unfortunately. You can try the Fauxbar Lite extension, which can be triggered directly in the Omnibar by typing
f <your query goes here>. It claims to use the AwesomeBar algorithm to search history, bookmarks, and open tabs, and it hasn't failed me so far after months of use!
Fauxbar will also add an icon to your toolbar; clicking it will bring up a page with an AwesomeBar clone. If you ever want to replace Chrome's New Tab page with the Fauxbar page, get the full extension instead.