Actually, I'll add pukunui's stuff here -- as usual it's great:
01-04 Mage Eyes: You can automatically sense the presence of magic within 30 feet of you. If you sense any magic in this way, you can use your action to see a faint aura around any visible creature or object in the area that bears magic (as per the detect magic spell). If you have proficiency in the Arcana skill, you can also determine the relevant school of magic, if any.
05-08 Cat's Grace: Your facial features are slightly feline, and you have a habit of purring like a cat when you're happy. In addition, you can use Dexterity (Acrobatics) to climb, and when you fall 20 feet or fewer, you take no damage and land on your feet.
09-12 Contortionist: Your body is extremely flexible and stretchy. You are considered to be squeezing only if moving through a space less than 1 foot in diameter, and you can spend 5 feet of movement to automatically escape from nonmagical restraints, such as manacles or a creature that has you grappled. In addition, as a bonus action, you can extend your reach to 10 feet for the next melee attack you make before the end of your turn.
13-16 Make Like A Tree: Your skin has a mottled, bark-like texture, and you always seem to have leaves or twigs in your hair. In addition, you can use your action to cause your flesh to become as tough as wood for up to 1 hour, or until your concentration ends. As long as you are not wearing heavy armor, you gain a +2 bonus to AC for the duration (as per the barkskin spell). In addition, once per day, you can use your action to cast the tree shape spell.
17-20 Shifter: As an action, you can grow claws or fangs or some other natural weapon of your choice. Your natural weapon deals 1d6 bludgeoning, piercing, or slashing damage, as appropriate, and you are considered proficient with it. Your natural weapon counts as magical and lasts for 1 hour, or until your concentration ends.
21-24 Speak in Tongues: You understand any spoken language you hear, and when you speak, any creature that knows at least one language and can hear you understands what you say.
25-28 Darkvision: You gain darkvision out to 60 feet. If you already have darkvision because of your race, its range is doubled instead (eg. 60-foot darkvision becomes 120-foot darkvision).
29-32 Telekinetic: As an action, you are able to mentally manipulate objects as per the telekinesis spell.
33-36 "Earthbound": Your skin has a stony hue to it and feels rough to the touch; you no longer have any hair; and when you speak, there is a faint sound like the roar of an avalanche in your voice. In addition, you have resistance to bludgeoning and piercing damage from nonmagical weapons, and you have advantage on saving throws against being petrified. You can also cast the meld into stone spell once per day.
37-40 Flamekissed: Your skin has a reddish hue to it and feels warm to the touch; your hair has turned a fiery red and waves in the air like flames; and when you speak, there is a faint sound like a raging wildfire in your voice. In addition, you have resistance to fire and heat-induced exhaustion, and you also know the hurl flame [a refluffed version of sacred flame] cantrip.
41-44 Icebound: Your skin is pale and feels cold to the touch; your hair has turned bluish-white and always has a thin layer of frost on it; and when you speak, there is a faint sound like a whirling snowstorm in your voice. In addition, you have resistance to cold damage and cold-induced exhaustion, and you also know the ray of frost cantrip.
45-48 Stormtouched: Tiny, harmless sparks dance around your body as you move, and any hair on your head always stands on end as though statically charged. When you speak, there is a faint sound like the rumble of thunder in your voice. In addition, you have resistance to lightning and thunder damage, and you also know the shocking grasp cantrip.
49-52 "Waterborne": Your skin has a faint blue-green tinge to it and is moist to the touch; your hair is perpetually damp; your fingers and toes are webbed; and when you speak, there is a faint sound like crashing waves in your voice. In addition, you have can breathe underwater, and you can swim at your normal land speed. As an action, you can also fill an open container within 30 feet of you with up to 2 gallons of clean water; alternatively, you can create a 30-foot cube of rain within the same range, extinguishing any exposed flames in the area (as per the create or destroy water spell).
53-56 Door Lord: At seemingly random times, you catch brief glimpses of mystic doorways leading to other places, usually at the periphery of your vision. In addition, as an action, you can create a portal through which you can teleport to any other spot within 500 feet. It can be a place you can see, one you can visualize, or one you can describe by stating distance and direction (as per the dimension door spell). You can bring along objects as long as their weight doesn't exceed what you can carry. If you would arrive in a place already occupied by an object or creature, you take 4d6 force damage and fail to teleport. You can use this ability a number of times per day equal to your proficiency bonus.
[Inspired by the eponymous creature in the Adventure Time episode "What Was Missing".]
57-60 "Trackless": As an action, you can grant yourself the ability to move through any type of natural terrain as per the pass without trace spell. You can also move across any liquid surface as per the water walk spell. While this ability is active, it is impossible to track you except by magical means. The effects last for 1 hour, or until your concentration ends.
61-64 Green Thumb: You have an affinity for plants. You can instinctively sense whether a plant is harmful or beneficial. In addition, once per day, you can use your action to cause grasping weeds and vines to sprout from the ground in a 20-foot square centered on you for up to 1 minute, or until your concentration ends (as per the entangle spell). For the duration, the ground in the area is considered difficult terrain. When the effect ends, the conjured plants wilt away. Once per day, you can spend 8 hours communing with nature to enrich the land around you (as per the plant growth spell). Afterwards, all the plants within 150 feet of your location yield twice as much food when harvested for one year.
65-68 Skinchanger: You can use your action to touch the mind of a willing beast within 30 feet of you, gaining the ability to see through the beast's eyes and hear through its ears. You can continue to do so until you use your action to stop. While perceiving through the beast's senses, you gain the benefits of any special senses it possessed, but you are blind and deaf to your own surroundings.
[Inspired by the wargs in Game of Thrones.]
69-72 Defy Gravity: As an action, you can walk on walls and ceilings as per the spider climb spell.
73-76 Fadewalker: As an action, you can cause yourself to randomly blink in and out of the Ethereal Plane for up to 1 minute, or until you choose to end the effect (no action required; as per the blink spell). At the end of each of your turns, roll a d20. On a roll of 11 or higher, you vanish from your current plane of exist and appear in the Ethereal Plane. The effect fails and your action is wasted if you were already on that plane. At the start of your next turn, you return in a space of your choice within 10 feet of the space from which you vanished. Unless you have magic that can reach across planes, you can affect and be affected only by creatures and objects in the Ethereal Plane while you are there. If you roll a 1 on the d20, roll again. If you roll another 1, you are lost in the Ethereal Plane for 2d6 hours. At the end of that time, you reappear as described above.
77-80 Spell Reflection: Whenever a spell directly targets you, and you successfully save against it or it misses you, you can use your reaction to choose another creature you can see within 30 feet of you to be the target of the spell instead. This ability does not function against spells that affect an area, such as fireball or cone of cold.
81-84 Regeneration: At the start of each of your turns, if you have fewer hit points than your normal hit point maximum, you regain a number of hit points equal to twice your proficiency bonus. This trait ceases to function when you drop to 0 hit points.
85-88 Beast Whisperer: You are able to comprehend and communicate with beasts as per the speak with animals spell. In addition, once per day you can use your action to magically compel a Tiny beast to deliver a message for you as per the animal messenger spell.
89-92 Seven League Stride: As an action, you can gain the benefit of either the longstrider spell or the jump spell. You can freely use these effects a combined number of times equal to your Constitution modifier. Any usages beyond that place undue strain on your body, requiring you to succeed on a Constitution saving throw with a DC of 10 + 1 per extra use or suffer one level of exhaustion.
93-96 Fade Away: Your body is slightly transparent. In addition, you can use your action to fade away completely, becoming invisible--along with anything you are wearing or carrying--for up to 1 hour, or until you attack, cast a spell, or end your concentration (as per the invisibility spell).
97-00 Telepath: You can communicate telepathically with any creature within 100 feet of you that understands a language.
FYI: The trait names in quotation marks are temporary; hopefully I'll come up with some better names for them at some point.
