What prompted R2-D2 to wake up at this point in “The Force Awakens?”

The current belief is that Rey's presence is what prompted R2-D2 to awake from low power mode.

When he's revealed to be at the Resistance base, the only character that's not there is Rey because she's been taken hostage by Kylo Ren. At the end of the film she's now back at Resistance headquarters, and her presence seems to be what spurns R2 to wake up.

The running suspicion is that Rey is related to Luke in some way. Remember that C-3PO tells BB-8 that R2-D2 went into low power mode not long after Master Luke left after Kylo betrayed him and destroyed the new Jedi Order he was attempting to build.

However, without official word that her presence is what caused R2 to wake up, we'll likely have to wait for the inevitable sequels to know for certain.

From an Entertainment Weekly article with J.J. Abrams titled, “J.J. Abrams explains R2-D2's closing scene in Star Wars: The Force Awakens”:

“BB-8 comes up and says something to him, which is basically, ‘I’ve got this piece of a map, do you happen to have the rest?’” Abrams said. “The idea was, R2 who has been all over the galaxy, is still in his coma, but he hears this. And it triggers something that would ultimately wake him up.”

The director acknowledges that R2’s sudden “awakening” at the end was designed to be an emotional storytelling utility: “While it may seem, you know, completely lucky and an easy way out, at that point in the movie, when you’ve lost a person, desperately, and somebody you hopefully care about is unconscious, you want someone to return.”

So for those let wondering: BB-8’s earlier question rattles around inside R2’s dome for a while. Those old astromechs must just take a while to boot up again.

Then as the movie draws to a close, our old friend finally comes back – and leads us to another one.

They will probably explain a little more/better in the next movie.

Pure theory crafting.

Luke wants Rey to find him, when the time is right.

I base this theory on Rey being Luke's daughter. Luke leaves her because things go bad with Kylo, but he understands that Rey is the only way to counter Kylo's power. He leaves her on Jakku and flies away (why? to protect her and to keep Kylo busy).

His plan is to be found when Rey is older, after some "tests", so he can be sure she won't turn to the dark side (like Kylo did). He brings his old lightsaber to Maz and instruct R2D2 to await for Rey's coming. He could have given him some instructions like "wait for an heroic action from Rey", "she must have the lightsaber", "Kylo must be unable to track her", something else...

Even simpler, R2D2 could be in direct contact with Luke, sending him updates on the situation. Luke would then use a remote activation when he's confident enough that his daughter won't follow the knights.

How did R2D2 know that Rey is Rey? He knew how to recognize Kenoby in Episode IV, so he must have some proximity sensor to do that. He already met Rey when she was a child, so he has her data in memory.

Of course these are all just theories and I'm probably not the best person to make a theory but this is what I think. I think that Luke (probably using the force) woke R2 up. I think that Luke felt that Han had died (which is why I think Luke looked so angry at the end) and that made him realized that he needed to come out of hiding and fight Kylo-Ren, so he made R2 wake up and show them where to find him. I think that Luke had hope that Kylo would turn back to the light side, but after killing Han, Luke probably assumes that there is no turning back for him now. Who know?

In Rey's premonition she saw Luke touching R2D2. It could mean that her premonition got something to do with R2D2, that she is the only one that could activate it for her to find Luke Skywalker. As you can see in the end, Luke is not surprise that Rey found her as if he is expecting her to come.

It is not the presence of the lightsabre because Fin had the lightsabre earlier & had not given it Rey. Earlier the lightsabre, a Skywalker heirloom beckoned Rey, R2-D2, also a Skywalker heirloom, may have the same connection to sense her presence, I would guess she is a skywalker & both her parents, whoever they are will play large roles in her destiny. I think Luke placed the Map minus the missing piece in R2 & put the droid in low power until a Rey arrived, which he must have planned for since he placed the missing piece of the map in such close proxity to her. As far as Kylo goes, I theorize that Leia must bring him back from the Dark side, since she was the one that "sent" him away. I think it fits into the whole balance of the force thing. Just my two cents. Feel free to rip it apart now.