As examples, here's what I've thought of. Didn't put much thought into balance issues. If you have any ideas for perks which would be better suited for the particular Daedra (or ideas for Perks for Aedra-worshipping players), I'd love to hear them. Commentary in Spoilers, usually explaining things that aren't immediately clear.
Azura’s Blessing: All Abilities increase +25% faster during the twilight hours; 5% of saluaging Soul Gems after use; Starting Spell: Starlight
Spoiler
Starlight should be familiar to players of Oblivion as a low-level Light spell.
Boethiah’s Blessing: All Lies twice as likely to succeed; May attempt Lies a second time if first attempt fails; 5% Chance to gain Sneak Attack bonus while in open combat; Starting Spell: Frenzy
Spoiler
Chose Frenzy for Boethiah for the ability to cause foes to betray each other.
Clavicus Vile’s Blessing: 10% better prices on all items; Persuasion increases Speech twice as much as usual; Starting Spell: Charm
Spoiler
This one was difficult. Ideally the game would have some gambling options and Clavicus Vile would boost your chances at that. Especially interested in ideas for this one.
Hermaeus Mora’s Blessing: Skill books provide a temporary +10 bonus for 4 hours after reading, in addition to the usual permanent +1 bonus; Boosted skill increases +10% faster for duration; +50% better prices on books; Starting Spell: Clairvoyance
Hircine’s Blessing: Predatory beasts will not attack unless provoked; Eating meat provides twice as much Health or Stamina; +50% better prices on animal parts. Starting Spell: Call Wolf
Jyggalag’s Blessing: +10% Magic Resistance; +10% Damage Resistance; Starting Spell: Dispel
Spoiler
Another tough one. I figured Jyggalag would keep his chosen "unchanging" so this is what I came up with.
Malacath’s Blessing: Power attacks cost -10% Stamina, inflict +10% Damage; Starting Spell: Feather
Spoiler
Gave Malacath Feather because he appreciates strength, but didn't want to give his followers a Fortify Strength spell (assuming Strength were to make a return) with the permanent damage boost to Power attacks.
Mehrunes Dagon’s Blessing: Weapon Abilities and Destruction increase +5% faster; +25% Damage to guards and soldiers; Starting Spell: Flame
Spoiler
Not sure about this one. Attacking guards/soldiers might be too limited, but would be handy if the game involved fighting for/against the Thalmor/Empire quite a bit. Otherwise I don't fight the guards very often. Not sure about other people.
Mephala’s Blessing: Sneak Attacks do twice as much damage; Poisoned weapons last twice as long; Starting Spell: Muffle
Spoiler
This was the last one I did. Had a hard time sorting out Boethiah/Mephala/Nocturnal and figuring out which Daedra gave what to their followers.
Meridia’s Blessing: +25% Damage to Undead; Soul Gems provide +10% Magicka for recharging; Starting Spell: Detect Life
Molag Bal’s Blessing: May peacefully interact with Vampires without becoming one; Intimidation and Bribe increases Speech twice as much as usual, but Persuation increases Speech only half as much; Bribes cost -50% less; Starting Spell: Command
Spoiler
I'd like to see a vampire faction (or factions) in future games; this perk would let the player do those quests without actually becoming a vampire. More likely than not, some of those vampires would offer to turn you into a vampire if you desired, too.
Namira’s Blessing: All Abilities increase +10% in Caves and Mines; Eating insects and mushrooms doubles positive effects and ignores negative effects; Starting Spell: Extinguish
Spoiler
I loved the 'spell' Namira's Shroud from Oblivion. Extinguish is basically that. It would extinguish torches, campfires, and similar sources of light.
Nocturnal’s Blessing: All Abilities increase +10% faster at night; all Bounties gained at night reduced by 25%; 5% chance that new Bounty will be dropped if never caught by Guards before next sunset. Starting Spell: Call Raven
Spoiler
Call Raven would 'summon' a raven to you. It perches on your shoulder until commanded (read: activated). You can command the raven to fly around, letting you look through its eyes and scout ahead. It can also pick up small items and deliver them to you, but has a limited carrying capacity.
Peryite’s Blessing: +25% Disease resistance; 25% chance to infect attacker in melee range with random disease, increase to 50% if currently infected with a disease; diseases automatically purged at 24 hours; Starting Spell: Infection
Spoiler
Infection would be a Destruction spell that infects your target with a disease and drains various stats depending on which disease it is.
Sanguine’s Blessing: Alcohol and drugs have +50% positive effects and -50% negative effects; all effects last twice as long; Starting Spell: Merriment
Spoiler
Merriment causes a brief fit of laughter. This effectively paralyzes the target but will immediately terminate if the target is attacked.
Sheogorath’s Blessing: Bizarre and unexpected events seem to stalk the player; Starting Spell: Madness
Spoiler
This was the perk that started this idea. Basically it mimics the Fallout: New Vegas perk, Wild Wasteland, which populates setting with unusual events, easter eggs, and other craziness. Madness, in fine Sheogorath tradition, would be a randfom effect spell, potentially casting either Frenzy, Calm, or Fear on a target.
Vaermina’s Blessing: Never gains Rested bonuses; +10 Magicka per hour slept, decreasing by -10 Magicka for every hour thereafter. Starting Spell: Dreamcatcher
Spoiler
Dreamcatcher is a Restore (maybe Absorb...?) Magicka spell that must be cast on sleeping targets and cannot be cast on the same target more than once a night.