Personally, I'd use something like this. A fairly minimalist and hacked up version based on the Sorcerer. Honestly, while there's a good argument to be made about having psionic abilities divorced entirely from magic, the 5e designers have committed pretty hard to the idea that pretty much all supernatural powers should be mechanically represented by spells, simply to increase interchangeability and avoid having to support another subsystem in future sourcebooks.
Level 1: choose any 1 of Int, Wis, Cha to be your spellcasting ability. One of your saving throw proficiencies is your spellcasting ability, the other is Con. Psionics require no components. Perhaps a 1/long rest ability to meditate to regain expended sorcery points and/or spell slots, cos the sorcerer is arguably a bit underpowered on that front right now. Power reserve is handled by the standard sorcerer ability of being able to swap sorcery points for spell slots.
A really quick and dirty mechanism for separating psionics from magic. Non-psionic characters trying to dispel psionic effects roll at disadvantage, and vice versa, non-psionic characters using Detect Magic to detect psionics need to pass a Perception check against the spell save DC, and vice versa.
A spell list separate from the regular sorcerer spell list. Lose the flashy evocations, conjurations, transmutations, instead gain various divination, enchantment, buffing spells from the bard, druid, cleric lists.
POSSIBLY some sort of 'tell', in place of components? Perhaps when you manifest a psionic spell, there's a perceptible effect in a 15 foot radius of you, maybe the temperature drops or people hear whispering in their heads, or a sense of mental pressure, or maybe your eyes glow. This is more flavour than anything else though.
Subclasses grant additional spells known at level 1, a bit like domain spells. At 6th level, psions can cast their special subclass spells that are not the highest level they can cast, with a slot one lower than normally required. At 14th level, they can maintain concentration on two spells simultaneously as long as both spells are from their subclass list.
Subclasses:
- mindbender: gets all the good telepathic/psychic damage spells, and phantasmals. Additional features: free telepathy, psychic damage resistance.
- shipfloater (for want of a better name - this is what dark sun called telekinetics, and I like it as a not-too-science-y psionic term even though its very setting-specific): telekinetic and projectile spells like Catapault, plus stuff like Slow/Fly etc which plausibly have a telekinetic origin. Bonus feature is free Mage Hand, a reaction to grant resistance against incoming missile weapons, and any telekinetic effects have no visible effect if desired.
- firestarter: bunch of fire spells. Bonus fire resistance and Elemental Adept
- seer: every divination spell, from every class, plus some abjuration etc spells that can be interpreted as effects of precognition or something. Bonus feature, luck rerolls like a divination wizard
- shaper: a bunch of transmutation and abjuration spells with a 'self only' restriction added - enlarge, polymorph, stoneskin, cure wounds, barkskin etc. Bonus feature would be granting self darkvision or natural weapons or water breathing etc, and possibly extra HP and some sort of option for making melee attacks using spellcasting ability, cos this is a combat-focused subclass and the sorcerer is a bit fragile for that sort of thing otherwise. Besides, if I'm gimping their transmutations to be self-only, it's only fair they get more powerful features in exchange.
I haven't even TRIED to balance this, but that's the shape of how I'd do it.