I often make the same search queries in Gmail (e.g. give me all the mails I sent to a specific mailing list). Unfortunately, these queries are not kept by Gmail and you have to type them everytime.
But I found a simple way to make Gmail remember them.
So, if you want to access Gmail queries directly from its search field history, you just have to search for them on Google (provided you have the Google cookie).
Since Gmail seach field kept the history of your Google searches, they will be available next time you search mails in Gmail.