Sona:
Adopted by a Demacian woman who gave her a magical instrument called etwahl.
Discovered the true power of the etwahl and joined the League.

Nami:
Her race was screwed because an ancient stone was losing its power, so she decided to help them by becoming the Tidecaller.
No matter what she could do, her race was still screwed, so she joined the League to seek the stone.

Twisted Fate:
Wanted magic.
Back-stabbed Graves for it and joined League afterwards.

Karma: Was a b*tch, helped ionia hold out by becomeing a stone faced b*tch.
Joined the league because potatoes.

Nunu

Dad dies
raised by yetis

Jax.

Jax OP.
Only uses lamp post, no longer OP, but still pretty awesome.

Draaaaaaaven

Once a Noxian executor, who put on shows
Then joined the League to do the same on the Fields of Justice