In 5e, the owl is often way more attractive than other familiars: between high-speed flight, darkvision, and flyby, it’s just really good at all the things you mostly want a familiar to be good at. I put the following together to try to level the playing field a bit, such that a player wanting to take a different animal as a familiar for roleplaying reasons won’t feel like it’s a bad mechanical choice.

It’s up to you whether you want all versions of the animal to have these abilities, but it’s definitely intended to be something that a familiar version of it gets.

Bat

Night Flier: The bat has advantage on Dexterity (Stealth) checks and attack rolls (to deliver touch spells) at night in dim light or darkness. Attackers have disadvantage on opportunity attacks against the bat in these circumstances.

Cat

Nimble Escape: The cat can take the Disengage or Hide action as a bonus action on each of its turns.

Crab

Big Claw, Little Claw: The crab may reroll a failed attack roll to deliver a touch spell, and may take the Help action as a bonus action on each of its turns (allowing it to effectively Help on two attacks per round).

Dog (Little Dog)

(Treat stats as Cat that doesn’t have a Climb speed, has Keen Hearing and Smell, and its attack is Bite instead of Claw.)

Pack Tactics: When the dog delivers touch spells, it has advantage on its attack roll if at least one of its allies is within 5 feet of the creature and the ally isn’t incapacitated.
Fierce Protector: While taking a short or long rest, if the dog succeeds at its Wisdom (Perception) check to notice an approaching threat, none of the rest of the party can be surprised.

Frog (Toad)

Lashing Tongue: The frog has 10 foot reach for delivering touch spells.
Hopping Defense: The frog may take the Dodge action as a bonus action on each of its turns.

Hawk

Swift Striker: The hawk may take the Dash action as a bonus action on each of its turns, and has advantage when delivering touch spells during any turn in which it’s used the Dash action.

Lizard

Detachable Tail: When the lizard is reduced to 0 hit points but not killed outright, it can drop to 1 hit point instead. It can’t use this feature again until it finishes a long rest.

Octopus

Clever Tentacles: An octopus can manipulate an item for its master with as much or more dexterity than human hands. When within 5 feet, it may grant the Help action on any of its master’s actions involving manual dexterity.

Owl

The owl is already really good.

Poisonous Snake

Venomous Delivery: When the snake successfully delivers a touch spell, it can also inject venom if the master desires. The target must make a DC 10 Constitution saving throw or take 2d4 poison damage.

Fish (Quipper)

(The fish’s Blood Frenzy applies when delivering touch spells.)

Rat

Streetwise: When inside a city, town, or building, the rat may take the Disengage or Hide action as a bonus action on each of its turns.
They’re Everywhere: When it is reduced to 0 hit points by a ranged attack within an area that has other rats, it is instead missed (the attack killed a different rat). The rat may only use this ability once per day in the same location (approximately within the same 100 feet).

Raven

Gregarious: In any location that has other crows or ravens, the raven exchanges gossip with others of its kind. The raven automatically becomes aware of threats the corvids know about, and can take 1d4 hours to make a Wisdom (Perception) check to discover more specific information for its master. The DC of the check depends on how easy it would be for local corvids to uncover the information.

Sea Horse

Surprising Speed: The sea horse can take the Dash, Disengage, or Dodge action as a bonus action on each of its turns.

Spider

Patient: If it is not moving, the spider gains advantage on Dexterity (Stealth) checks to hide.
Venomous Delivery: When the spider delivers a touch spell, it can also inject venom if the master desires. The target must make a DC 9 Constitution saving throw or take 1d4 poison damage.

Weasel

Nimble Escape: The weasel can take the Disengage or Hide action as a bonus action on each of its turns.
Weasel’s Intuition: The weasel can Help its master with Charisma (Deception) checks if it is within 5 feet.