On the advice of wiki user "Downtochill", I hooked up my Android to my laptop with a USB cable and went hunting for a root directory related to the game. Unfortunately, it wasn't as simple as finding a GAME directory. No, nothing ever is. I did find a frequently accessed directory with a very odd name:
Digging down into the folders I found several groups of Mobage and My Monster Rancher assets including player profile icons, Mobage coin icons, screen graphics, and so on. I concluded that ".ngmoco" was somehow meaningful to the Mobage people even though it meant nothing to me, so I deleted that directory. Then I restarted my phone and re-installed My Monster Rancher. There were a couple of weird messages about switching to wifi (I was already on wifi!) and "This game is still under development for your device" (scary! is this thing still in beta?) and of course I had to log in again. Then came an extended "Getting new content" delay and a really long "Downloading..." screen, but this time the "Downloading..." screen was active and the green bar was filling incrementally so I waited.
Miracle of miracles, it worked! I was back in the game with all of my item inventory, neighbors list, and monsters just as I had left them nearly a week ago.
Which is kind of odd, actually. I was offline nearly a week and nobody removed me from their neighbors list? I prune my neighbors list daily and anybody who doesn't log in once every three days or so gets cut. I get between five and ten neighbor requests every single day, so there is very little reason to keep someone as a neighbor if they aren't active. There are a few exceptions to this such as "depsky", "donna022786", and "vixen66666", because I regularly raid their freezers when combining low-level monsters. I'd keep them around if they were gone for months!
So, two lessons were learned from this experience:
1> every 90 days or so I will have to remove the game and the game directory
2> keep a well-stocked freezer and you'll have happy neighbors even if you get locked out for a week or so!