I am working on a new class, and thought a popular vote would be the best way to build it, and it's subclasses.

As the class will include an ability that a beast might have, I would like a vote of favorite abilities, or abilities that the beast might possess which isn't listed in their stat block.

For example, the octopus.  It's body structure allows it to mold it's body to fit through spaces that appear much smaller than it's body. This could be considered an ability. 
Ignore tight spaces - Mold your body to fit through spaces that normally only a being 1 size category smaller could fit through.

List the ideas in the following style:

Octopus - Underwater Camouflage - The octopus has advantage on Dexterity (Stealth) checks made while underwater.

What are your favorite beasts and beast abilities that you would love to have as a player character?

I have always been a fan of charge, pounce, and dive. They all work pretty similarly.

Charge/Pounce/Dive - if the creature moves at least X feet in a straight line before hitting with an attack, the target of that attack takes an extra (dice type and amount) damage and must make a strength save (DC = 8 + prof + str) or be knocked prone and the creature can make an additional attack (typically of a different type than the first one) against that target as a bonus action.

this is found on horses, giant eagles, tigers, and others. The first attack is usually a claw or gore attack and the bonus action one is most often a higher damage bite or hoof attack. 

Those are called “special traits,” and I’m a fan of Pack Tactics and Amphibious, as well as Keen Senses/Keen Sight/Keen Hearing/Keen Smell. Spider Climb and Standing Leap are cool too.

These are a good start.