OO fun!
I know some of this might not be exactly meshing with what others want, but it is
my wishlist.
#1: STOP calling them Psionics and make them "Psychic" like they always should have been.
#2: Simple system: Power points seems the best way to go. Really, you could already run a psionic with the sorcerer, as is. Though it would be a bit more limited than I'd like to see psychics. Just need a decent list of powers.
#3: Simple subclasses: I would go as simple as 3: Telepath, Telekinetic and Clairvoyant/Seer.
But a Psychic Warrior/Knight subclass for all the jedi-wannabes seems a foregone conclusion. So, I could live with 4 subclasses, I guess.
#4: What I do NOT need to see: A Psionic Fighter, a Psionic Rogue, a Psionic Mage, a Psionic Cleric, a Psionic Barbarian, a Psionic Bard, a Psionic...NnnnO! <swak> Bad developer. Put the tablet DOWN! I don'wannem.
#5: Please PLEASE kill the Soulknife/-blade/Psylocke ripoff. I know it'll never happen. But it's a "wish list."
#6: A simple optional subsystem for random psychic talents/abilities for [otherwise] "normal" classes. Like the 1e random chance to have psionics. So your could have a Ranger or Druid who is actually able to mentally or empathically speak with animals. Probably why they were geared in that direction/interest in animals/nature from a young age. A mage with only a random/slight touch of precognition. A cleric who can heal minds/psychic damage as well as bodies...again, likely gearing her toward a career/interest in healing to begin with. Etc... Random talents.
#7: [which legitimately should maybe go up the list, like to #2] PSYCHIC POWERS & ABILITIES! There will be overlap with magic spells, certainly. Particularly enchantment type stuff and some illusions. But I do NOT need to see psychic cellular rearranging, shapechanging, physical combat abilities [well, I guess the telekinetic would have something applicable here, but still], etc... I don't need a 1-to-1 with stuff other classes or types of magic. They are "Psionic" powers. Make them PSYCHIC! Make them MENTAL abilities!
I should be able to stun/knockout targets with bolts of psychic energy. I should be able to make guards think they are paralyzed or make myself invisible [influence their mind to not "see" me]. Communicate/call allies over great distances. I should be able to throw/grab/manipulate physical objects and block/redirect energy types with my telekinetic force fields. I should be able to scan an area for other sentient and psychically aware/capable minds. Send out my "mind's eye" to reconnoiter, find out where the MacGuffin is and keep tabs on the BBEG's location, so we can avoid it.
And, naturally, engage in mental/psychic battle with other psychically capable beings: fight off or exorcise possessing creatures/psyches, block my (and my companions') mind(s) from intrusion or control, battle mindflayers on the astral/mental/psychic plane [however it is fluffed] and shatter their minds, maybe counter Magic Jar and/or undo Feeblemind.
I think that's all I really want out of 5e psionics. As usual, I will probably just homebrew/continue using my own psychic powers system.