Most probably because Leia knows what Rey is, as does Luke when he sees the lightsaber.
I stand by my theory that Rey is a reincarnation of Anakin Skywalker, one that Luke had been waiting for a long time.
Here's how I think it'll play out. At some point after ROTJ, Anakin's ghost begins to feel drawn back to our world via a child. He tells Luke of this and prepares him for what that could mean for the Rebellion/Resistance. If Snoke got his hands on Rey/Anakin, the Dark Side would have the weapon they needed [again] to take back the galaxy. Instead, Anakin urges Luke to hide the child he's being drawn to, from everyone - even Luke himself so if Snoke captured him he wouldn't be able to pull her identity from his mind. Luke gives the task of hiding her to Lor San Tekka who drops her on an old Sith world (Jakku) and basically is the only person who knows how to find her (and Luke). Prior to vanishing, Anakin's ghost tells Luke they will see each other again and that Luke will know who he is when he sees him (her).
What neither Anakin nor Luke realize is that The Force has its own plans. Rey/Anakin's time on Jakku is penance for the atrocities Vader committed as Vader. And when said penance is done (TFA) its The Force that guides Rey/Anakin back to Luke via Finn, Han, etc.
There are clues to this in Rey's vision at Maz's and her fight with Kylo. "THE FORCE AWAKENS". More like ANAKIN AWAKENS inside Rey.
I think that's way too complicated. Besides, its the blood -- doesn't have to be Anakin's soul... it's his blood line.
I assumed Leia put her to task knowing who she is, but why would Rey go? I thought she was all about going home to Jakku. Without a good reason why she has to get Skywalker, it's just a bit of a logic jump that is punctuated by the whole cast waving goodbye to her when she sets out. It's a bit silly. Needed a moment, between Leia and Rey, where Leia presses a reason why she alone must go. Maybe Leia says: Luke can tell you about your family. That would be enough for her to go. But we should have seen that.