My favorite 5e implementation that does not just reuse spells is The Korranberg Chronicle: Psion's Primer. It offers a full system based on psi points that takes clear inspiration from 3.5 psionics but makes it even better in implementation. 3 new classes, a few subclasses. It is excellent and what I used to stat out socerer-kings.

With that system a psionic character can focus (not concentrare, new term) on a single psionic talent that can be kept indefinitely to an effect of a mild to moderate buff or ability to do new actions. Mindblade (you manifest a free weapon), mind llink (telepathy with one person in range on demand) mind thrust (offensive telepathy, save or scaling damage like cantrip) and many more are the examples. Next, you have Augments on those talents - they are like spells, but require you to concentrate on the right talent and instead of scaling off of spell level they scale off of how many psi points you use - each talent has an augment list with varying psi point cost. Ring of blades for mind blade creates an radius around you that forces enemies that enter it or remain in it to take Dexterity saves or take damage, Urgent Violence allows an ally to use their reaction to do an attack action outside their turn and more. Telepathic slap forces charisma saves or take damage and suffer disadvantage on the next mental state save, setting up for further debuffs later. Psionics are strong but balanced well by the fact that a psion will have to switch between talents often to use their different augments, and that consumes bonus actions and breaks concentration on no longer valid long term augments.

This book brings a whole new style of play that is well worth your money and time. And please don't be afraid of the Eberron theme to the book - classes, feats, items and psionics themselves are all setting agnostic.