New Star Wars film is titled Star Wars: The Last Jedi. In this movie we can see "Kylo" has Force powers, "Rey" has Force powers at the end and Princess Leia also has Force abilities. Luke Skywalker is already a Jedi.
So, who is the "Last Jedi" referenced in the movie title? Is it:
Luke Skywalker
Kylo Ren
Rey
Princess Leia
Though, the title itself is quite ambiguous, but we can go through the movie plot and conclude it.
Princess Leia is sensitive to the Force, but she is not a Jedi. She didn't get training as Jedi.
Kylo Ren was trained as Jedi, but he was seduced by the dark side of the Force, so he is not a Jedi.
Rey was just learning the ways of the Force, but wasn't under actual Jedi training. She might be a Jedi in future, but it's speculation for now.
When master Yoda died in Return of the Jedi, Luke was the last Jedi left. Here is the conversation between Yoda and Luke.
Yoda: Luke...Luke...Do not...Do not underestimate the powers of the Emperor, or suffer your father's fate, you will. Luke, when gone am I (cough), the last of the Jedi will you be. Luke, the Force runs strong in your family. Pass on what you have learned, Luke... (with great effort) There is...another...Sky...Sky...walker.
When Kylo Ren fights with Luke near the end of The Last Jedi, they have following conversation,
Kylo Ren: And when I kill you, I will have killed the last Jedi.
Luke Skywalker: Amazing.
Every word of what you just said...
was wrong.
The Rebellion is reborn today.
The war...
is just beginning.
And I will not be the last Jedi.
And the opening crawls of The Force Awakens says,
Luke Skywalker has vanished. In his absence, the sinister FIRST ORDER has risen from the ashes of the Empire and will not rest until Skywalker, the last Jedi, has been destroyed.
So, it is clear that the title is referring to Luke.
Luke Skywalker is referred to as "the last jedi" by the opening crawl of Episode VII: The Force Awakens:
Luke Skywalker has vanished. In his absence, the sinister FIRST ORDER has risen from the ashes of the Empire and will not rest until Skywalker, the last Jedi, has been destroyed.
The text of the Episode VII opening crawl can be found on this wiki page: "Opening crawl", Star Wars wikia
Rian Johnson, the writer and director, seems to indicate that it is indeed Luke, though he seemed hesitant to do so in April 2017
During a chat with ABC News in April, Johnson wouldn’t confirm the title as a reference to Luke. “Well, if you say so,” he said to the suggestion. “I’m gonna take your word for it. They say in The Force Awakens that he’s going to find the last Jedi temple, and Luke is the last Jedi.”
But another interview in September shows more certainty:
“It’s in the opening crawl of The Force Awakens,” he clarified. “Luke Skywalker, right now, is the last Jedi..."
At the start of the movie, Luke Skywalker is The Last Jedi. He is the only Jedi-trained individual in the Galaxy who has not fallen to the dark side.
At the end of the movie, Rey has everything she needs to become a Jedi and train a new generation. She has the will, the ability to stand up against temptation, and the Jedi tomes stolen before Yoda burned the shrine down (beside the blankets in the Falcon). There are children throughout the galaxy inspired by the tales of Luke and Rey and companions, including one who force-pulls a broom before striking a pose.
I suspect the real reason of the movie's title is the paradox it evokes:
The Last Jedi has died, but he will not be The Last Jedi.
As of the end of The Force Awakens, we believed Luke Skywalker to be the last remaining Jedi in the galaxy. Even the opening crawl of The Last Jedi calls him this:
Luke Skywalker has vanished. In his absence, the sinister FIRST ORDER has risen from the ashes of the Empire and will not rest until Skywalker, the last Jedi, has been destroyed.
However, the name of the film is a twist, playing into the surprising (?) events of the film's end:
Luke dies, and in voiceover indicates that he is not the last Jedi.
Camerawork and events happening elsewhere indicate that, instead:
Rey is.
Whether you believe this is good storytelling and/or actually makes any logical sense is quite another thing. :)
Well. Even though the answers so far make sense, in all foreign translations into languages which differentiate between plural and singular "The Last Jedi" was translated into the plural form, indicating at least two (arguably Luke and Rey).
The fact that it happened on every possible instance indicates that this was a deliberate top-down decision, though it could just be miscommunication between writers and marketing.
