In the new Star Wars Movie,
Rey is searching for Luke Skywalker to give him his lightsaber.
But as you can see in Episode VI, Luke built his new green one with another handle. Didn't she just know that he already got another one?
She was reaching out to a man who had shut himself away from the galaxy...
...by giving him something that he has a personal connection to — his long-lost lightsaber.
Even if she was aware of a fine detail such as his newer green lightsaber (and there's no reason she would know something like that when it's clearly established in The Force Awakens that the Jedi are more or less legend and myth by that point), giving him a lightsaber so that he can have a lightsaber is not the point.
Giving him something to sway him after shutting himself away for years as a hermit is the point:
(Official novelization, Chapter XVIII)
First of all, Rey offering Luke her weapon shows that she means no harm. But there is much more significance to offering that particular weapon.
The lightsaber belonged to Luke's father before him. It is the only memento he ever had of Anakin Skywalker. It is an object of deep personal significance, and (so far as we know) he thought it was lost forever in Cloud City after his father cut off his hand. He will at the very least be interested to see it again.
Furthermore, while Rey could have stumbled across the lightsaber by blind luck, Luke is likely to believe that she was guided to it by the Force. The fact that Rey has the lightsaber is an indication of her importance and great destiny, which is another reason for her to show it to him.
Rey knows at least some of the lightsaber's history (having been told it on screen by Maz Kanata). She might also know that Luke has a new green lightsaber; Han, Leia, Chewbacca, and C-3P0 are all aware of it and could have told her off camera. But Luke's new lightsaber is irrelevant to her reasons for offering him the old one.
Finally, Rey would have been searching for Luke whether she had his lightsaber or not; she wants to be trained as a Jedi, and it is strongly hinted that Luke knows something about her past and family background.
Rey gave Luke the lightsaber, but she was not:
She had it, and she was looking for him, and it was his, but they don't flow in the order you're suggesting. She was looking for him for training; having the light saber was incidental.
In the scene with Rey and Han on the Millennium Falcon, it's made quite clear that she's aware of the legend of Luke Skywalker, but it's also just that - a legend. His place in history was before Rey was even born.
The viewing audience may be aware of small details like that, but these are stories carried by word of mouth over thousands of parsec by (potentially) hundreds of thousands of people. The telephone effect makes it far more likely that any information she has is wrong and/or incomplete.
Because it belongs to him.
His father built it; he inherited it. Despite Kylo Ren's insistence that it belongs to him as some sort of heir to Vader's legacy, (which the light saber itself refuted by responding to Rey rather than to him when they both reached out with the Force to take it) that light saber belonged to Anakin Skywalker. Vader's light saber was red, like other Sith light sabers.
In returning his property to him, Rey is showing him respect, which is an absolute requirement if she is to become his padawan. He may well lend one of his light sabers (whether one of the two we know of, or some other he or one of his students has since constructed) back to her to use during her training, and may even at some point convey ownership to her outright, but that must follow her acknowledgement of his clear title to his father's light saber.
Besides, her training will not be complete until she builds her own, which will be hers in a way no transfer of ownership can ever truly convey.