Here (in West Virginia) "mailbox money" means money that you do not work for, as in royalties from an oil or gas well on you property. Every month a check from the Oil & Gas Drilling Company arrives in your mailbox and you didn't have to lift a finger to earn it. Folks are mighty envious of anyone getting "mailbox money"!
In reply to @FumbleFingers
Yes, as a surveyor, I know many people whose minerals (under their farms) have been leased to an O&G Co. The O&G Co. comes in, leases your minerals, drills a well, and you get a check in the mail typically for 1/8th royalties of the production. That's "mailbox money" because you haven't done a thing other than walk down to the mailbox once a month to get your check.
The ad you found (don't invest! don't do it!) wouldn't be considered mailbox money out here because it's a working interest and you are putting up the money to drill the well. You're probably right, it is a tax dodge... because a working interest in a well means that company is gonna charge you for every expense in the leasing, drilling, upkeep and management of that well; you're footing their entire cost before you see any possible profit.
And in further conversation with @FumbleFingers: We may have stumbled onto a word in transition across the rural-urban / 20thC-21st C divide. Out here "mailbox money" is likely to come from others using your land. In urban and more modern settings "mailbox money" would likely come from something intellectual or artistic you did once that people still pay to use. (Would be nice, eh?) Since someone (or their Grandpa) bought (or homesteaded) the land, work was done back in the day, same as with a clever slogan, artistic image or app. But methinks the books for sale on Google might be blurring the line between "mailbox money" and "get rich quick without doing any real work" schemes!