Probably a combination of looks, some base for characterization (as far as some established boundaries and sketches, as well as lots of room for headcanon.
People have been taking minor characters from a work and putting them into their own works forever (something like Rosencrantz and Guildnstern Are Dead came to mind), so it's nothing new.
I believe one of the major reasons for Abelas's popularity in particular is that we don't know much about him. The other reason is his appearance, particularly his body. I find him interesting and fairly attractive (despite the nose that rivals mine in prominence--I am more into faces than bodies personally). His codex entries give some extra insight into him, and it seems he's a rather tragic character. It might be nice to see him work through his issues and try to move past his grief. Let's face it, someone whose grief defines them so much at this point that they've actually named themselves "Sorrow," well... that's deep. Maybe some of it is just that people want to see him happy. I can't speak for everyone, but he is in my fic and I've got another one planned that will involve him (albeit I can't say he'll end up happy).
To me, though, there is a lot of appeal in how plot-involved he might be if Abelas were to show up (companion or not) in DA4. Think about it: He's an ancient elf who has clung to what remains for millennia. A chance appears to restore what was lost and rebuild. How could he not take it? But on the other hand, many of the Sentinels were born after the Veil, and are shemlen themselves. If he takes this chance, he may be betraying them. Would he betray them even though they are in effect his people (despite being shemlen)?
Ameridan... well, he's fascinating too. I wish we could've spent more time with him and learned more, but I don't think I'd have tried to romance him--even if he was quite attractive for an older man. Reason? Telana. He'd be in love with her for all time. At least with his death they could finally be together again.
What I do not understand is people going out of their way on both sides to bash each other for liking Abelas, or liking Solas. There's room for both. Nor do I grasp why people have to attack Solas fans for liking him after Trespasser, or really anyone who attacks anyone else based on their preference. For one, they aren't real people. Solas isn't destroying the REAL world. And for two... even if I don't find Iron Bull or Blackwall attractive, there are people who do. There are people who adore each and every character. It's disrespectful to go on the attack because someone likes a character you don't. It's childish, even.