Hard to go for just one favourite. Since it's really all about the games, I couldn't possibly pin it on just one console.
Excluding all those (great) Mario and Sonic and failed mascot games:
SNES for Uniracers, Secret of Mana, Chrono Trigger, Mother/Earthbound and plenty more.
GENESIS for the mad amount of outstanding sidescrolling/platforming fare - Super Shinobi/Revenge of Shinobi, Contra: Hard Corps, Kujaku Oh, Magical Hat no Buttobi Tabo! Daibōken, and some unique (action) RPG trips - Crusader of Centy, Soleil, etc.
PSX/PSOONE for all the meanwhile dead franchises that got optical-media-based gaming started. I still have not to this day seen anything like Rollcage on any current "Next-Gen" console. Simple little games that load fast and blast your senses with motion and colors and blur and a splendid selection of music tracks you probably wouldn't have stumbled over yourself.
The PS2 didn't amaze me as much, even though we own way too many games for this one to completely dismiss it. Still, when we go PlayStation retro, it's usually PSX stuff. MediEvil!
The NeoGeo for its instant display of boatloads of hand-pixelled character animation art. The Last Blade and Samurai Spirits franchises are still best enjoyed without the loading time lag of optical storage. Oh, and even though some just have to buy their EA soccer fix year after year, everyone here still enjoys a session of Super Sidekicks 2. Easy enough for the kids, but also compatible with beer-guzzling grown male specimens.
I tried some of these titles from the various retro selection sections of modern consoles, but they are mostly ported/presented in such a half-assed way that they're just no fun at all.