Before you read any more of this, I want you to know that if you have not seen The Force Awakens. I would also be remiss if I did not say that while the following post represents my own personal thoughts on the subject matter, some of them might be true and therefore spoilers.
The underlying plot of Star Wars Episode VII: The Force Awakens is that everyone is trying to find where Luke Skywalker fled to after the death of his Jedi students at the hands of the Knights of Ren, led by, of course, Kylo Ren. This was an especially devastating blow to Luke as Kylo Ren was once known as Ben Solo- the son of Han Solo and General Princess Leia Organa (Skywalker) Solo, his nephew.
In a flashback, we see what looks like that event from the perspective of Rey, whose parentage is a source of vigorous debate among fans, but the consensus seems to be that she is a Skywalker and most likely the daughter of Luke and some unrevealed woman. This theory had gained a lot of traction based on the skills Rey has not only as a hand-to-hand combatant but as a pilot (something both Luke and his father Anakin had in common, most likely due to their affinity with the Force) as well as her rapid development in the use of the Force which had been seen only twice before int he films, and only once with someone of similar age - Anakin and Luke, respectively.
At the end of the film, once the main conflict has been solved and the map leading to Luke is now complete, Rey and Chewbacca travel to the planet on which Luke has been living and he turns to her as she reaches out to him with the lightsaber he lost in the duel with his father. He has a very peculiar look on his face as he turns from what certainly looks like a headstone for an unknown grave.
Speculation on this has been rampant. First off, is this actually a grave? There are a lot of folks on both sides of this argument; some are sure it's a grave and others are of the mind that it's just a random outcropping of rock.
For this, we are going to assume that it is a grave. And not just any grave, but the grave of someone who is close to Luke, part of his family. Maybe not literal family, but perhaps part of the Jedi family at the very least. And I don't mean a long-lost Jedi because there are some people who are of the opinion that Luke is on a world that is home to some long-lost Jedi Temple, maybe even the first Jedi Temple. I disagree.
There are several people that those who believe the Luke is standing before a grave believe that grave is for. I will list them with a brief reason.
1.Luke's Wife - For those who believe that Luke and an unknown woman (Jyn Erso?) are Rey's parents this is the evidence they point to. They believe that Luke had to bury his wife here after Kylo Ren's betrayal at Luke's reformed Jedi Academy after Snoke corrupted Ben Solo's mind and made him turn towards the Dark Side and betray his family.
2. Darth Vader's Remains- As we saw in The Force Awakens Kylo Ren has the melted remains of his grandfather, Darth Vader's helmet. Perhaps Luke had taken the cybernetic parts that survived the funeral pyre and buried them here on this planet, the site of the first Jedi Temple but left the helmet behind as a way of representing Vader's turn away from the Dark Side and return to the Light.
3. It's One Of Many Graves of His Students Killed by Kylo Ren - Perhaps this isn't the site of the very first Jedi Temple, perhaps this is where Luke's Academy was located? You'd think that it might not make sense for him to hide there, but remember how he was hidden from Vader in the first place- on Vader's home planet. It's the old Purloined Letter trick- the best way to hide something is right out in the open where people aren't likely to look for it. Of course that would require you to disregard the fact that EVERYONE was trying to find Luke for one reason or another, and it took a two-part map to find him.
Those are some interesting thoughts, and any of them could turn out to be true, but based on how he looks at Rey at the end of the film, I believe that it's actually...wait for it...REY'S GRAVE!
Yes, it's Rey's grave. I know it sounds insane, but hear me out. Luke wasn't around for the slaughter of his students at the Academy, because had he been there, he doubtless would have dispatched Kylo and the Knights of Ren. What do we see in Rey's lightsaber flashback? We see her in a field full of bodies, and someone moving to strike her down, but they're stopped when Kylo drives his lightsaber through that individual's back much the way he did to Han Solo. The very next memory we see is her being left with Unkar Plutt on Jakku. Add this to the reaction Kylo has at being told that a girl helped the BB unit escape with the map to Luke and it becomes clear; Luke thought she was dead and went into hiding because of his perceived failures as a teacher.
Because Luke didn't have her body, the grave is symbolic- like a sailor lost at sea. He has been on this planet meditating on the Force and trying to figure out what to do when he senses her land close by. He still doesn't believe it, not until he turns around and looks at her for the first time in at least a decade. His expression changes from incredulity to understanding; Kylo Ren is the one who hid Rey on Jakku. He made sure no one could find her because he didn't want anything bad to happen to her, because they are related somehow. How? Well that's a theory for another day.
So that's my thought; Luke was mourning Rey (who may or may not be his daughter but was certainly a powerful Force user with incredible potential, like Luke and Anakin before him) and that's whose grave we see in the closing shot of The Force Awakens. Do you agree? Disagree? I'd love to hear your thoughts!