Ok first up I've some questions about the scenario.
You say well stocked with food and refreshments. What food, what drink... even water becomes stagnant when bottled and on a shelf for more than a few months. Stagnant water isn't going to be good to drink. Rice can go off despite being a "dried food" and against most people's belief in the fiction of media stereotypes. Canned food, has a shelf life of between 6 months and about 2 years at best before it will go off. Military Emergency Rations may last longer than that (several years in fact) but without a source of fresh water and the ability to heat that water to boiling point (so you can clean the water up of bacteria and germs) means you're going to die of dysentery, dehydration or starvation pretty quick.
Secondly, we're completely alone without hope of rescue? Without hope of other people finding us? Being joined by loved ones or even just another person no matter if you hated them? The reason why I ask is because hope of rescue is what keeps people in these survivalist situations alive. Hope of seeing their loved ones, or even just another person. Watch Castaway, Wilson the ball was created through the need for a human companion. It's to do with the whole "we're social creatures". Without hope of being rescued or leaving that place ever, you'd probably commit suicide pretty quickly. In which case the game you choose for the rest of your life doesn't really matter it's going to be a pretty short life.
Third up, the environment. You said Desert island; tropical desert or tundra? Hot or Cold? How large is the island? What types of weather do we need to look out for? What tools/equipment do you have/could you make to build a suitable shelter? I'm asking this because having proven that the initial food supplies will run out or go off. I'm thinking of how we'd actually live a life providing there was reason to do so. You're looking at the Minecraft situation here, chucked in to the wilderness with nothing but the clothes on your back what can you do to survive? In reality we'd probably all die. However, say you have a starting tool allowing you to make some basics you'll have a chance to live Minecraft. As such choosing Minecraft may be a bit redundant as the game for the rest of your life. Though with multiplayer servers allowed there's an important social aspect to the game which cannot be overlooked. It's also readily accessible for many people including non-technical people such as most of my family.
Assuming however, we have hope of rescue/escape, decent natural resources and expansion options, food supply and fresh water, clement weather and the skills to actually survive in this "dimensional rift" you're looking at games what would improve your life on a social scale. However, with all the tasks you'd have to do daily to keep surviving you're not actually going to have all that much time to spend gaming! As such time consuming titles like World of Warcraft may not be as good as you'd hoped.
Having given this some clear thought I'm tempted to say something as simple as Words With Friends could well be the best choice. It'd exercise your mind plus you can play friends/family or strangers. The fact it has a messenger program built in too is pretty sweet. Socially your life probably wouldn't be as bad. You'd also be intellectually challenging yourself which gives you better odds of survival as you'll become smarter. If we disallowed the internet then The Sims would be great for this as your sim goes off to make a social life, could however get a little depressing when you remember you are not your sim. More time consuming games such as Skyrim, Fallout, WoW, Final Fantasy are good options in so far as they will take a long time to complete which with your now extremely limited time span for gaming you're going to have months/years of gameplay to keep your evenings distracted. However, Elder Scrolls with exception of one title, Mass Effect with exception of 3, Fallout and Final Fantasy (with exception of 11 and 14) are all single player games only. A Multiplayer game would have a higher social impact on your life and therefore give you better hope to keep going. As such I find myself leaning towards multiplayer games I enjoy for long periods of time. That my friends enjoy for long periods of time and would be happy playing with me for a lot of time, long past the lifespan of the average console. Which also means I find myself looking at PC games not consoles. As proven by the passing of time and several generations of Xbox and Playstation; eventually network support for the old consoles is pulled. That being said the games that come immediately to mind are mostly console based. Destiny, Halo (series), Mass Effect 3, Minecraft, GTA V, Saints Row 3, all games where the most amount of my time has been spent on my Xbox 360. Worse still, with exception of Halo 1, Mass Effect 3 and Minecraft; some of these games aren't even on PC so take themselves out of the running. Taking that previous sentence in to consideration. I'm not sure what specific title I'd go with. I'll need to get back to you later with that.