Answer by Sean Burnside:
While I agree with Levi that Yoda was tired and would have preferred to have been left alone, his statement to Obi-Wan’s force spirit in ROTJ leads me to believe that they did have a plan. I don’t think Leia was part of that plan. I believe they didn’t and wouldn’t have trained her because she wasn’t suitable for the training.
Keep in mind that Yoda was a traditionalist when it came to accepting Padawans. He was even reluctant (with good reason, as was soon obvious) to accept Anakin, who was a mere child. Leia being a very mature, resolute young adult would have been completely unacceptable to him. I believe the only reason he chose Luke was because Luke -while headstrong and reckless- was still ultimately an innocent and open to change.
Leia, much like her future husband Han, was very worldly and very shaped by the worlds she lived in. Leia was a pragmatist, not a romantic and would have been very resitant to the training that a Jedi requires. Her royal upbringing had trained her of her social obligations and expectations. She was literally raised to rule Alderaan, to become a senator, to be a leader. Jedi might lead, but it is only by example, not as officers or beauracrats. Her desire to lead, to be a leader, to be important and significant were part of who she was and are anethma to being a properly trained Jedi, who are humble by nature.
Beyond that, I believe that Yoda and Obi-Wan (especially Obi-Wan) had a secret plan for Luke. Lucas has said that the sextology was the story of Anakin Skywaker, of his rise and fall and ultimate redemption. With that in mind, I suspect that Luke wasn’t intended to defeat the Emperor and Vader, but to redeem his father. On the whole, Luke (at that time) was a relatively weak Jedi. He was untrained and had insufficent experience connecting to the Force in order to be a truly great Jedi. Since Vader was an exceptional swordman and strong in the Force (pre-cybernetics, at least) and Darth Sidious was incredibly strong in the Dark Side of the Force, Luke would be unable to face them on even footing in battle.
Luke’s strength was his romanticism and optimism. He truly believed his father was still fundamentally good, that there was something worth saving and something capable of saving. His ultimate task wasn’t to defeat the Emperor, but to rob the Emperor of his greatest agent and have that agent -who was incredibly strong- in turn defeat his former master. I would say that the Jedi Masters felt this might be the best (and possibly only) hope they had.
So, why not Leia?
She hated Darth Vader. To her, Vader was the epitome of the Galactic Empire’s evil. I’m sure she knew about the Emperor and hated him, but she especially loathed Darth Vader. He was irrevocably linked to the obliteration of her entire civilzation and that is something that is not easy to forgive. Leia despised Vader completely. Worse, she had no respect for his history or his abilities. To her, he was simply a vicious, cruel thug. You can see this in conversations in the movie (although it is far more prevalent in the Extended Universe novels).
Furthermore, Leia already had a father. Senator Organa was a wise, just man. By all accounts, Leia loved her adopted parents and family. Now she learns that a monster she never knew -who was at least partially responsible for the obliteration of everything she knew, of the parents she loved dearly- is her true father. No, Leia would consider that nothing more than a biological fluke. She would never recognize his parentage. By contrast, Luke -who never knew a father and only had an uncle and aunt who no doubt feared what he might become- eventually welcomed the idea that Darth Vader was his father, primarilly because he believed Vader was redeemable.
So, let’s suppose that Yoda did train Leia and sent her to face her to face Vader and the Emperor. Assuming that she was as skilled / lucky as Luke, do we believe she would have shown Vader mercy? Would she have stayed her hand and denounced everything that the Emperor had planned and stood for? Luke’s faith in his father and rejection of the Emperor’s plans and power was what convinced Vader that he had a choice. Leia was too filled with hate towards Vader to spare him and would most likely have slain her father and irrevocably fallen to the Dark Side. I think Yoda and Kenobi suspected this and would have used her only as a last resort.