Is there course of action that I could take that could get this attacker to stop attempting to hack my girlfriend's email account?
Under recovery options in Google Account, just add your cellphone number. Whenever someone requests password recovery, the first thing Google does is send you an SMS (Text) message containing the code that you have to enter for recovering. The hacker cannot get this code any other way.
In the recovery options in Google Account, do not use a pre-specified question. Use a question of your own, something that only you can understand. A typical question would be : HOIW? Now the hacker would have no idea about HOIW? Just do not go wild and create a question /answer that even you forget later.
Also make your password is at least 12 character long. Use a password generator like KeePass
to generate a random password for you containing lower case, upper case, numerals and special characters. You can keep this password encrypted in the KeePass.
As an added security measure, change your password after every few months just in case.