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My girlfriend currently works as an intern in California on a J-1 visa. The internship (and thus the visa) ends Jan 31 next year.

If she were to quit her job before the end of the internship, what would happen to her immigration status? Does it depend on the action of her employer?

More specifically, the idea is stay for another 1-2 months to work on her job applications and participate in online courses, while living in our apartment and traveling for a bit (my visa is completely independent of her's).

I'm aware of the 4-week grace period after ending the job, but ideally we were looking to spend another 6-8 weeks. Furthermore, leaving and re-entering the US (on the visa waiver program) is not an option due to the expensive flights.

From the State Department's J1 Visa FAQ:

What will happen to my Exchange Visitor Visa status if my exchange visitor program sponsor terminates my exchange visitor program?

If your sponsor terminates your participation in their exchange program for just cause, the sponsor will enter this information into SEVIS and you will be expected to depart the United States immediately. You will not be entitled to the post-completion 30-day period because you did not successfully complete your program.

This is explained in more detail on this page under "termination."

So no, she would not be able to stay in the US after quitting her job and would be expected to depart promptly. The 30-day grace period only comes at the end of the program. Note that some J-1 visas are run through sponsorship programs (an organization separate from the employing company), so you'd want to check with them: some will allow you to get a different job instead of leaving immediately.

Note that the expiration date on the visa is only relevant to when she can enter the United States. She can only remain in the United States as long as she is "in status" with the immigration authorities. If she is out of status (such as if she stops doing her internship), it doesn't matter how long her visa is valid for.

If she quits her job, or her program comes to an end in any other way, she will no longer be in status. The employer is required to report (via SEVIS) that the program has come to an end.

After this occurs, the visa holder has a 30-day grace period to leave the country. If they stay longer than this it counts as an overstay.

You say that leaving and re-entering isn't an option, but a flight to Mexico or Canada is likely to be cheaper than the other option, which is to change status within the US. If she does leave and re-enter (or change to B visitor status), she should make sure she's only planning to do activities permitted under the VWP.