Our latest Lady of the Legion is Kara Zor-El of Krypton - or at least of a big chunk of what remained of Krypton after it exploded - more commonly known as Supergirl:
Kara's father, Zor-El - it was a Kryptonian custom for females to be identified by their first name and their father's full name - was the brother of Jor-El. Unlike the rest of the populace, when Jor-El predicted Krypton's destruction, Zor-El believed him, and undertook an effort to enclose his hometown of Argo City in an indestructible dome, which allowed Argo to survive the explosion of Krypton.
Adrift in space, the bedrock upon which Argo rested underwent the chain reaction that created Kryptonite, and so for the people of Argo it was out of the frying pan and into the fire, as they'd survived the destruction of their homeworld only to have the remnant on which they were living begin slowly killing them.
Much like his brother had managed to rocket his son, Kal-El - better known as Superman - to Earth, Zor-El and his wife Alura decided to save the life of their daughter, Kara, by sending her away to live with her cousin.
For years, Supergirl served as Superman's "secret weapon," keeping her existence hidden from the world.
(And, as I've mentioned before, making Kara incredibly neurotic in the process. He even made her live in an orphanage, and required that she sabotage any efforts by loving parents to adopt her, for fear that any adoptive parents might find out her secret. Yeah.)
During that period of secrecy, the Legion traveled back in time to give Supergirl the opportunity to try out for membership.
Kara passed with flying colors, and was on the fast track to membership, but in the course of her tryout she'd been exposed to Red Kryptonite, a variety of the irradiated remains of Krypton that has unpredictable - but temporary - affects on Kryptonians. In this case, it caused Supergirl to age several years, becoming a Superwoman.
Because the Legion bylaws stated that no member could be older than 19 Earth years old, and also because, as Chris Sims will tell you, the Legionnaires are often a bunch of dicks, she was rejected for membership.
Eventually, she did get into the Legion, serving with distinction.
In the 1981 mini-series Secrets of the Legion of Super-Heroes, which I just recently reacquired, it was "revealed" (read: retconned) that the Legion had actually approached Supergirl to offer her membership before they had recruited Superboy.
The reason? They were too intimidated by Superboy's legendary reputation, and wanted to make a practice run with his also-ran cousin.
Like I said, a bunch of dicks.
Still, it wasn't all bad. While in the Legion, Kara had something of a romance with one Querl Dox of Colu, a super-genius who went by the name Brainiac 5. The "descendant" of Brainiac, one of history's greatest villains and one of Superman's deadliest enemies, Brainiac 5 devoted his life in the Legion to trying to rehabilitate the name "Brainiac."
Given that he created a computer that ended up turning Triplicate Girl into Duo Damsel by killing one of her duplicates, and just generally having more brains than sense, I'd say he didn't do the greatest job of it...