No, not a badge or an achievement...
But what if every time you mourned, in addition to getting favor for that giant, that particular giant told you a snippet of a narrative?
Each giant would have it's own complete tale to tell, but to get all the stories, you'd have to mourn a lot of gravestones for each giant. And as you complete each story, you can unlock a little "Summer's Day"-esque area for each giant, which you could reminisce at gravestones.
(This idea started as a joke, "What's the story, mourning glory?" but I actually think this could be a fun little addition.)