This is how I'd do it...
Enter scene: Rey stands before Luke Skywalker, her arm outstretched towards him holding his long lost lightsaber....
Luke: "Now that's a lightsaber I've not seen in a long time. A long time" (Luke rubs his mechanical hand with his left hand).
He reaches out slowly and Rey hands the Skywalker Family Lightsaber to him.
Luke: "My Father wanted me to have this when I was old enough, but my Uncle wouldn't allow it." (CRACK-HISSS!!!! the lightsaber springs to life! The light illuminating Rey's face and eyes)
Luke: "This is the weapon of a Jedi Knight. How did you come by it?"
Rey: "It... called to me, but I want nothing to do with it. It's yours, you keep it."
Luke: (Deactivates the lightsaber) "And what would I have need of it for?"
Rey: "What? You're needed to help defeat The First Order. General Organa-Solo sent me to deliver it to you so that you could fight them!"
Luke: "I have forseen a great many things. The Force is strong in my Family. My Father had it, I have it, my sister has it..."(Luke looks down at the ground) "her son has it. My..." (Luke looks back up at Rey)... "I have no need of this lightsaber."
(The ground shakes. Giant boulders elevate behind Luke and what appears to be a headstone cracks in half. Luke jerks his head to see the headstone fall over, a wave of emotion crosses his face, then snaps to calm and the boulders crash to the ground.)
Rey: " How can you say that? You're the Last Jedi! You have to join the fight!"
Luke: " No, there is another way. Always two there are, a Master, and an Apprentice."
Rey looks on confusedly.
Luke: "You are to learn the ways of the Force if you are to accompany me to this conflict. And you will need this..." (He stretches out his mechanical hand, still holding the lightsaber in it)
Luke: " I want you to have it."
I think that's way too many quote references from 30 years of lost time for Luke to directly quote from the OT, haha! I'm very interested to know HIS story of what happened during all that time lost in exile. What was he doing after the events of Kylo and the Knights wiping out the people he was training? He had to have been there for a deep purpose. As devastating of what happened to him trying to grow the Jedi order again, I don't picture Luke as one to just hide.
Even when the odds were against him in ESB he went in alone to try and help his friends. Sure he was younger (and more foolish?) then, he was also brave. In Return of the Jedi, he stood up to Jabba ALONE while his friends were not in the position to help him in the palace. Certainly a good Jedi such as Luke or Obi-Wan have the capacity to handle themselves.
What I want to know is, with as good as he is, how was Kylo able to deceive Luke and him not know it? Maybe only certain types can sense things such as Yoda, who must not have seen it either because he didn't drop by to give the heads up.
Yoda: "Luke.... Dude.... Betray you, Ben will."
Luke: "What are you talking about, Yoda? Ben is with you in the afterlife. Go away, little old creepy monster."
Yoda: "But....."
Luke: "Not hearing it. Happy April Fools Day to you, too..."
{10 minutes later the slaughter}
Luke: "Ooooohhhhhhh.... THAT Ben....."
Yoda: "You dumb-a**. To exile you must go."
There had to be some very dark power that allowed Kylo to do what he did and put Luke in check. He should have seen it.
On the note of Kylo Ren... I think many underestimate him with all these whiny claims. Anakin was whiny before he made his final turn. We only know Kylo in his younger time making his way through the First Order and not later on as he's matured (the way many know him as Vader). And give the guy a little break. He grew up in the almost complete absence of his parents. That can screw a person up pretty bad to not have the guidance and love from a Mom and Dad. This is his life, he's obviously trying to find purpose. He's damaged, angry, strong, and uncertain of where he is. It makes him a very dangerous character and interesting as well. I'm very curious to see if he makes a turn back to the good side. He obviously knows Snoke is taking advantage of him. I could tell his mind was rolling and almost questioning himself while Han was trying to tell him he was being used. Even though his Dad wasn't there much for him, he wouldn't lie to his own son, and I think even Kylo/Ben knows that.
Anyway, back to Luke and answering the question.... Even given all Luke has been miserable over dealing with his failure and loss, I can't see him throwing out the lightsaber. He's strong with the force, but not even Obi-Wan never kept his saber at home that I ever knew. He still needs it for protection just in case.
And that's a lot of garb he's wearing to conceal where a lightsaber could be held on a belt.