We can pick the consoles as they are, and what each model includes. That means full backwards compatibility (for the consoles/models that do have such) and region lock if there is such, yes. You also can have every possible add-on for the console, as long as it's from the official manufacturer of the console.

You can choose the given console's model with bigger memory cards, or bigger hard drives if you want, but you can not hardware mod them, because you don't have modchips on the island. However since you have internet there, you can soft mod some of the consoles that can be softmoded, such as PS3. From there you can unlock the region, but again, it depends on the console itself and if there is soft mod available, and if that allow the region lock.

My favorite: N64 (my 1996 charcoal gray model, which still performs perfectly). This carried me through middle and high school. So many formative experiences on this thing. So many happy memories of playing Mario Party, Star Fox, Smash Bros., Mario Kart, GoldenEye, and Perfect Dark with friends.

The best: Switch (OLED) or PS2 (slim). I can't choose.

The one I'd take to a desert island: Switch 2 (launch model). Thanks to NSO, it has games from NES, GB, SNES, Genesis, GBA, N64, and GCN. Thanks to backward compatibility and ports, it has games from WiiU and Switch. And, of course, Switch 2 games.

I'm just going to ignore the whole soft modding / reliability aspects.

PS5 is probably what I'd choose to have the whole PS5+PS4 libraries. Xbox Series X would probably have been my pick if it had full BC with 360/XB and not just select titles, 4 generations in one device is huge.

A launch PS3 is very close also though to have PS1+PS2 as well.

Launch model PS3 Fat that has the ability to play PS1/PS2 and PS3 games. Would have it be a brand new unit off ebay for $2000, and would replace all caps and thermals. Also would have someone do whatever crazy PS3 mods exist to improve the cooling.

Since NES is childhood, and since I have a billion games for it, it's gotta be that. I can play SMB 3 or Mega Man 1-6, or Ninja Gaiden, or Zelda 1, or Tecmo Bowl ad nauseam for eternity and they just don't get old. Can't say the same for any 3D games, even the best ones like OOT.

Probably the PS2 because it has the original Deus Ex, and a gadzillion of other great games, you can't beat the PS2 library.

Ay, another PS5 choice.

PS4/PS5 library combined is indeed one of the best in gaming. There's even tons of old games by themselves or in collections that you can play. 

I agree that if Xbox Series X had full or near full BC with Xbox and Xbox 360 plus the next two gens it already has that would be pretty high as well. 

Such a inspiring thread.
It's tough, I would like to play crash bandicoot 1 on a lonely island,harmony( first on is set on an island).

It takes a lot of time (and calmness) to complete it...!

However, based on that i would have to choose either ps1 or NS (the trilogy), while 3ds is my favorite console (something well known here 😅).
3ds all the way, playing Mario party hard mode or 3D Land perhaps.

Toss-up between the NES & SNES. It's really a hard pick. A plurality of my all-time favorite games are on the SNES, including A Link to the Past, Super Metroid, Final Fantasy IV, and Super Mario World. One the other hand, the NES had a very large and solid library as well, including my #1 favorite game of all time, that being Super Mario Bros. 3, plus it had Mega Man 1-6.