I agree that explaining the trip as "volunteering on a farm" is inviting trouble -- the "temporary visitor" permit you get under the Japanese visa waiver program does not clearly cover that sort of activity and Japanese immigration officials are not very confident in their critical thinking ability, so this young lady would seriously run the risk of ending up in the holding tank for a while as more senior officials decide what visa category she falls under. I think they would ultimately let her in as a "temporary visitor" as long as the work is unpaid, but it isn't a slam dunk.

"Staying with a friend" (who just happens to live on a farm) for up to 90 days is permitted, and if the visit can be legitimately characterized as such, there should be no problems.

My understanding is that for Americans, the extension of the standard 90-day entry permit to six months is only available if you demonstrate a compelling/humanitarian need to be in the country for that additional period -- for instance, if you are injured/ill while in the country and need some more time to recover.

I agree with the suggestion to talk to your local Japanese consulate about whether you can go visa-less to Japan. (If possible, ask them by email so that if you get a favorable response, you can carry a copy to Japan with you to support your position at immigration...)