3 years after MGS4, MGS3 still remains my favorite in the series (MGS3 > MGS4 > MGS > MGS2). The HD update in wide aspect ratio has been a treat/gift.
As for why it remains my favorite, the story was one of the better villain origin stories in gaming, fully fleshing out the character of Big Boss with a fantastic ending sequence fully explaining his motives and the source of his disenchantment with the concept of country/nations. It also had one of the more impressive, well developed "villains" in The Boss.
While it was a deviation from MGS, and a huge deviation from MGS2, which took the "future warfare" concept into the realm of near fantasy, I really appreciated the "lo-fi" 60s world of MGS3 which was deliberately designed to show the evolution of technology in the later games.
Plus, as an origin story, it works best as a standalone game that doesn't really require knowledge of the other MGS to be effective; it's just a lot more entertaining and the story beats have a greater emotional impact if one is familiar with the other games.
As a toy, with emphasis on the whole "play" concept, I just had a lot of fun running around the jungle collecting flora and fauna and eating it, or just hanging out in the jungle. For some reason it's still hugely satisfying, even in a post Crysis gaming world and what that game did for true open world jungle environments.
Color commentary from the Codec, Paramedic's in particular was pretty entertaining as well. The camo system was a bit clunky, but... 60s tech; no octocamo or optic camo yet. But it was a major play mechanic that both worked and added something rarely explored in games.
Lastly, it's probably the simplest, least convoluted story of the MGS series even though it has more than its share of WTF moments. Of course, it wouldn't be a MGS game without those.
"He's attacking me with bees?! Oh, right; it's an MGS game, so I'll accept it."