Oops, I meant to type “House Darry” instead of “House Dayne”… Dyslexia, the gift that keeps on giving. Sorry for that. It just House DARRY seems to have lost a lot for their loyalty to House Targaryen and that chapter in AGOT where Ned stays near there. Being blood kinsmen would help to explain it while House Blackwood doesn’t seem to be that close to the House Targaryen at any point in the narrative despite a fairly recent marriage alliance to not just any member, but the King himself.

Oh. 

Well, the Darrys are also an ancient House (one of the original First Men houses that managed to survive the Andals), and their shtick is the defense of the status quo no matter what – the first mention we have of them is fighting a losing battle against the Andals during the Andal invasion of the Riverlands, then we have them fighting on the losing side in defense of House Teague during the Battle of Six Kings, then getting almost wiped out by Aemond and Vhagar during the Dance, and on and on.  

This tendency has been encapsulated in song:

“And there he stood with sword in hand,
the last of Darry’s ten…
And red the grass beneath his feet,
and red his banners bright,
and red the glow of the setting sun
that bathed him in its light,
“Come on, come on,” the great lord called,
“my sword is hungry still.”)
And with a cry of savage rage,
They swarmed across the rill.”

Deremond isn’t just presented as a doughty fighter; he’s explicitly the last of ten Darrys who have fought and died upon the field, and he’s refusing to cede the field to a numerically superior enemy in favor of dying a heroicly stupid death. 

So I don’t think you need blood ties to explain their loyalty – for the Darrys, it’s enough that Aerys II was the king, Ser Jonothor was his Kingsguard, and Ser Willem his master-of-arms (even if only to piss off Tywin).