Found this comment on Reddit regarding Nami's reason for joining the League.

"The stone lasts 100 years, they lend her the stone and as long as she fights in the league they won't take it back. She's fighting for the league in exchange for saving her people.
At least that's how I would tell it."

There. It's not some amazing piece of storytelling but at least it makes some sort of sense. Much more than "she's on a quest to save her people but decided to stop and fight people in a League for no apparent reason".

Not sure who downvoted you, must be the redditor mentality of "I disagree, must downvote without thinking!" ... Back on topic.

I personally disagree, it ruins the interaction between Nami and Diana which is implied to occur in Nami's lore. While more explanations would be nice, the one that is offered would actually be worse then the present one.

I don't really care about Nami as a character too much so I never took interest in her lore but a better albeit longer explanation could be that she looks towards the league as a way of finding the powerful people on runeterra who would actually know where she might find the stone.

There is probably some explanatory lore. Of course Riot's writers know of it, but they won't tell us cause revealing any more lore than a champion's background is blasphemy.