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Making money off of it wouldn't be too hard, actually. Skins based off of key parts of the storyline is the first thing that comes to mind. A lot can be said just by looking at a skin, and people who like the story mode would definitely buy skins based on epic parts of the story. Think about it - you play through the storyline and then play a classic match or something, someone rolls in with a skin from a part of the storyline that you loved. You would instantly connect with that person. You'd have stuff to talk about and you'd likely work better in the match because you've bonded over something.
Additional maps/missions for the story mode could be something Riot could also sell. If the original story mode (which could be free) is very well-made and popular, people would be willing to pay for more of such content. Maybe have side missions/side storylines be purchasable as well.
Last, if we link tournament-style story events (like Iona vs. Noxus) to this, it could generate just as much publicity and money as the pro-tournaments. If you got some of the pro players to agree to play in these events, you'd not only draw the tournament crowd, but the lore and story-driven crowd as well.