So aberrations seem to be focused on two things:
1) alien nature
2) magic that originates from the mind
A player race based on Aberrations should focus on those things above all else... What about something like:
Xenokin
Xenokin are a race born of blood from beyond the known world.
Appearance: Xenokin look overall humanoid, but each has features that make them distinctly alien. Roll on the following table (or choose) for your alien feature:
Alien Features
1) One single eye, with vertical lids, in the center of the forehead.
2) A lamprey-like mouth surrounded by writhing tentacles.
3) Shimmering fish scales for skin.
4) A wide mouth full of sharp teeth.
5) Long, boneless limbs ending in tentacle-like fingers.
6) Skin the color of a night sky, in which glimmers distant stars.
Xenokin Features
Ability Score Improvement: Choose +2 / +1
Alien Mind: Other creatures have a difficult time understanding the workings of your mind. Creatures suffer Disadvantage when attempting to make Insight Checks against your character.
Psionic Adaptation: Your powerful brain is trained to adapt to and resist magic. When you fail a Saving Throw against a magic effect, you may choose to reroll with Advantage. You may use this feature once per Short Rest.
Shifting Form of Chaos: Choose one of the following features for your Xenomorph. After a long rest, roll a d20. On a roll of 1 or 20, choose a new feature.
Eyes of Black Starlight: You gain Darkvision (60 ft), and do not suffer Disadvantage when using Perception in Dim Light. You can see invisible creatures and objects.
Strength of the Ancients: You gain a climb speed equal to half your walking speed. You have advantage on Athletics checks made to grapple opponents.
Gifts of the Deep: You gain a swim speed equal to your walking speed, and may breathe underwater.
Hunger of the Endless Void: When you succeed on a Saving Throw, you absorb some of the magic or kinetic energy of the effect. As a reaction, you may spend a Hit Die to heal. If you are at maximum Hit Points, you gain Temporary Hit Points equal to the total rolled.
Psionic Power: You learn one Cantrip from the Enchantment, Illusion, or Divination schools. Your casting ability is Intelligence, Wisdom, or Charisma.
Boundless Levitation: You gain a fly speed equal to your walking speed. This flight is powered by your mind.
That's what I came up with off the top of my mind. Alien, Psionic, and inspired by the Aberrations of the Monster Manual.