As we know, Regulus is almost always grumpy, and even when he is happy, he still looks like an angry cat. I feel like whenever he would be miserable a little too much, Barty would always try to make him feel better.
It started in the middle of first year, when Regulus was going through a depression episode (he was upset about Sirius ignoring him), and Barty wanted nothing more than to make Regulus happy. He tried multiple things. He got him candy, hexed Sirius, and pranked the marauders, but nothing was helping.
One day, when Barty came into the dorm, he saw Regulus' bed with all the curtains closed around it and a silencing charm around it. He immediately knew something was up and asked Regulus to let him sit on the bed with him. After some convincing, Regulus opened the curtains and dragged Barty onto his bed, and quickly closed the curtains for privacy. Barty could tell Regulus had been crying because of his red eyes and stuffy nose. It wasn't even a minute later when Regulus started to break down. Barty, of course, was a bit freaked out since Regulus never showed any emotion at all (the only way he knew he was depressed was because of how distracted Regulus was and because he wasn't reading late at night but just staring at the wall). As Regulus sobbed, he started to cry about how worthless he felt now that Sirius wasn't around him and that he would never be good enough. After listening to Regulus cry his heart out, Barty finally got over his shock and did the first thing that he could think of. He hugged him.
Regulus froze for a minute out of shock because the only person who had ever hugged him was Sirius more than a year ago, but after getting over his shock, he leaned into Barty and continued to cry. Even after he had stopped, he still cuddled with Barty on his bed and refused to let go (not that Barty was complaining or trying to get him off of him) until they heard Evan's footsteps coming towards their dorm.
Ever since then, whenever Regulus was even the slightest bit upset, him and Barty would cuddle for hours on end until they were forced to let go because of classes or other people to close to their hiding spots for their liking.