What I found important / useful so far:
Resists - Good / Effective
Mana Regen - Necessary
HP Regen - Good / Effective
Special Moves - Fun but eats mana waaaay too quickly
Skill cap increase - Not tested but I expect will be useful
Extra skills (Ninjitsu, Spellweaving, Necro etc) - Really quite cool but replaces Magery and no summons (shame)
HP - Really good but takes a lot of upgrade points
Stam - Good / Effective
Base damage - Good Effective
New nightmares seem to be 3 slot by default which could lead to 2 slot legacy nightmares becoming really valuable if they retain their 2-slot control requirement and upgrade from 2-3 slots instead of the new ones going from 3-4 slots (sorry to start the mare-rush if this is true).
Haven't tamed or tried much but so far (will update if I do any more more testing):
Shadow Wyrm (always wanted one which is why I am here lol) 5 slots but not as good / survivable as a GD and looks like it can't be upgraded. Still really cool though ... did I mention it was really cool ?
Nightmare (fresh tame so 3 slots) - Upgraded goes to 4 slots but not as good / effective / survivable as a Dog (Cu-Sidhe)
Tsuki Wolf - Upgrade from 3 > 4 slots. +Necro / Resists / Regen / Feint / HP. Good fun and surprisingly tanky
Rune Beetle - Upgrade from 3 > 4 slots. +Resist / Regen etc. Wanted to try one with Chiv or Spellweaving or Necro but you don't seem to be able to do that

Buffed 4 slot rune beetle is what you get basically without the damage output of Beetle+Mare but at least less pets to manage
Skree - Haha what a great little beastie. I'm guessing there is a few bugs still to be ironed out as I managed to make a Skree as strong as a GD. The rate these little guys cast at is hilarious. Certainly the most fun with a 5 slotter I've ever had !
I'm definitely not touching any of my old pets until the FOTM upgrades are well documented and tested.
Night