The last time we see River chronologically, she tells Johnny to take the money to finish the house and watch over "Anya." Anya is the lighthouse, so he kept his promise. And in spite of everything, she does Go Out with a Smile.
John: And what about you? River: ...Happy. I'll be happy.
Sadly, Johnny never learns why she made her blue and yellow rabbit. On the bright side, Johnny had the piano carried into her room to play one final song dedicated to her, which she gladly listens to.
Finally learning exactly why River made all those rabbits, especially the blue and yellow one. It was to remind him of his promise to meet her on the moon... which was surrounded by stars that Johnny and River noticed looked like a rabbit.
River's reaction when Johnny tells her that they may be able to buy the lighthouse in a not too far off future. This is one of the very few moments through the entire game when her emotions openly show and swell up so much that she just plainly throws herself into John's arms in overwhelming joy.
The dancing scene in the lighthouse. The bathos that happens when they trip over each other's feet make it more endearing.
In the memory when River and Johnny are riding horses. Even though John is initially hesitant when River is allowed to gallop on her own for the first time it quickly melts away when, and the player can see this if s/he looks closely enough, River laughs. Equine therapy for the win.
When Johnny was close to dying prematurely and Eva and Neil were still in his memories.
As heartbreaking as seeing the destruction of Johnny's memories of River was, it's equally heartwarming seeing them reconstructed as they go up in the rocket together.
When Johnny and River are reunited at NASA. Although this wasn't real, it shows that in his mind, no matter where she was, no matter when or how he meets her, he will always fall in love with her, and that despite their communication problems, she was irreplaceable, and he literally can't picture his life without her.
Some players found it unsettling that Johnny's memories were ultimately replaced by an alternate life. While this is a valid concern, it helps when you consider that even with the new memories, and even when desiring to go to the Moon, Johnny still loved River so much that he subconsciously brought her back. And his Dying Dream shows his true priorities; Johnny doesn't even get to make it all the way to the Moon before he dies, but his last memory is of him together with River, gazing at the Moon just like when they were kids. "If you're with me" indeed...
A very subtle one could be in the doctor's office after river has just been diagnosed with her "condition", when asked if they are a couple River states that they are in no way legally bound, either one of her quirks or perhaps her indirectly telling Johnny that he is free to leave her.
Of course he stays
Just before the credits roll, Johnny and River, in their child forms, reunite against a white background and Johnny hands the platypus to her before sitting down on the log next to her, strongly implying that they're Together in Death, regardless of the hardships they'd faced in life. Also, this is one of the rare occasions where River smiles.
Meta-example. The Steam version of the game has a lone achievement. It is called "Wish Granted", has the tooltip "send Johnny to the Moon", and a picture of Young River and Young Johnny as the thumbnail.
Eva trying to convince Neil to return to the Christmas party in the second minisode.
Eva: You do what you want; I'm heading inside. Just know there's a chair upstairs for you, whether you're coming or not.