Simple really. Maybe the main series of a franchise is not for you but a spinoff or series of spinoffs you enjoyed. A good example is Mana series from Final Fantasy. Persona from Shin Megami Tensei.  Mario Kart to Super Mario.

I can give an example some may not expect with one obvious one. I like the lore of Metal Gear but not into the gameplay of the main series. I do love some of the games that take place in the universe of Metal Gear.

Snatcher.

Policenauts

Zone of the Enders 2

Metal Gear Rising

Snatcher for those unaware is Kojima's Blade Runner but it does exist in the MG universe. It's not directly clear where in time. However, it does have a Metal Gear. Big Boss is here as well.

Policenauts is very much in the MG universe with the very first appearance of Meryl. Anyone who played the original MGS has heard part of the Policenauts theme with the intro of the game on the Konami logo.

ZOE2 is decades if not longer in the future and while it takes place in the universe. It has no direct connections.

MGR is pretty obvious.

I love Cyberpunk so Snatcher and Policenauts are a given. I like old adventure games as well. ZOE2 is just one of the best mecha and action games ever made that is still amazing. The fast-paced gameplay is almost PG like.  MGR is a PG game and one of its finest.

It's hard to imagine too many people that would qualify for this. If someone already dislikes a series, it's hard to imagine them liking a spinofff, because the traits they disliked will probably be in that spin-off. Maybe widen the parameters to 'series you didn't care about' or something broader.

To give an example, people who don't care for shooters wouldn't play Gears of War, but may like Gears Tactics because it's a tactics game instead. If people dislike Gears of War, it's probably because they don't like the overly masculine designs and gratuitous violence. In which case they still won't like Gears Tactics or any spinoff regardless.

So personally, I'd say Crash Nitro Kart for the original Xbox. I never owned Crash Bandicoot games growing up, but Crash Nitro Kart was perfect for Xbox which came with 4 controller slots (opposed to PS2 which required buying a multi-tap, and then buying another when you realised PS2 slim uses a different version), and so I played that a lot with family.

I'll aslo throw in World of Warcraft, since I never played the Warcraft games.

I don't like soccer in general (or don't care about sports in general) but I remember getting into Fifa Street with my cousin for a couple of days. Don't like Fifa in general, but the first game had a very arcade-like flow to it, si we played it quite a bit.

don't remember a case, closest would be enjoying persona without ever playing shin megami tensei.

A few examples:

Did not like X-Com and I thought the Rabbids games were only ok.  Mario + Rabbids is pretty fun though.

I have no interest in playing any Might and Magic game.  Heroes of Might and Magic 2 is one of my favorite games of all time.  (HoMM 3 is great too.)

Only played Skies of Arcadia for about an hour before my Dreamcast broke, but I didn't really care for it based on that.  I really, really love Valkyria Chronicles.

Honorable mention goes to the Dynasty Warriors games, which I do like, but are nowhere nearly as good as either Hyrule Warriors or Dynasty Tactics.

DQ heroes II but this was before I enjoyed DQXI.  

Not unusual to see spin-off of spin-off.

They share the same demons. Jack Frost is in them and some entires let you recruit demons in the part like SMT. Older Persona dungeons are similar. Until Persona 5 they were called Shin Megami Tensei: Persona

Metal gear rising, yep, me too
My girlfriend enjoys a lot pikmin adventure, of the nintendo land. But she hated pikmin.

Ahh, nice example. I wasn't really coming up with anything, but in this same respect, I don't care for sports games but I very much enjoyed and played a lot of the NBA Street games (or at least the PS2 versions, didn't play the others).