Now that's an oversimplification and this is a personal opinion - I like my games lean, mean and many options, few rules. Many of the psionic systems that I've taken the time to really read and grok end up following a similar pattern:
1. Figure out how powerful the psionics are (and in some systems, figure out how much psionics you can do for this level/day/combat.)
2. Figure out the attack/defense
3. Figure out if an attack/effect was successful (usually matching results from #1 and #2, comparing, and making dice rolls for saves and damage.)
4. Track points/power levels.
There are a lot of people who like that type of mini-game, I'm not one of them. Call it thick-skull, maybe I inhaled too many fumes from that "dope" when I was building model airplanes as a kid, but I like my in-game systems quick and easy.
That all being said, I do have a psionics system I do like, and it was written by Gary Gygax in 1975 in the magazine Strategic Review #1. In it, he described the Mind Flayer for the first time and it's Mind Blast power:
Intelligence Saving Throw at Range Effect of of Opponent 1-2" 3-4" 5-6" Mind Blast 3-4 19 19 17 Death 5-7 17 16 15 Coma, 3 days 8-10 15 14 13 Sleep, 1 hour 11-12 13 12 11 Stun, 3 turns 13-14 11 10 9 Confuse, 5 turns 15-16 9 8 7 Enrage, 7 turns 17 7 6 5 Feeblemind 18 5 4 3 Insanity, permanent Magic users add +1 to their saving throws, and clerics add +2. A Helm of Telepathy adds a +4 to saving throws, and when such saves are made the attacking Mind Flayer is stunned for 3 turns.
Now admittedly, to my own taste, there's a lot wrong with this chart, such as why someone with 18 INT would suffer insanity if they fail their save versus the 13/14 would just be confused, BUT it's the simplicity of the effect and approach. Attack. Save. Ouch or Haha! So let's expand this just a smidge.
1. Figure out if you have psionics - prereq is a 13+ INT. 1 in d30 gets you the freaky mind!
I do like Carcosa's system of using INT/WIS/CHA to determine likelihood, so I'll add that a WIS of 16+ and/or CHA of 16+ adds +1, cumulative. A person could have a 3 in 30 if they have high enough stats. 10% chance.
And yes, you can simulate a d30 with a d20+d10 but really... the d30 is so cool, this system will make you run out and buy one. Right? Guys?
2. You have 4 powers - Mind Blast (like the mind flaya' [although I may redo this to do some damage dice and save the effects for a failed save]), ESP (like the spell) and Charm Person (like the spell). You also have Telekinesis (like the spell). You can use the number of powers per day as your level/3 (rounding rules apply). So choose wisely at first level.
It was my thought that 10th level should be able to use all powers at least once, so I may allow 10th level 4 times since that's the cap on most classes. Maybe not, maybe those freaky high level mages should allow themselves some extra oomph, since we are talking about OD&D here...)
3. To attack, you declare like a spell and let your mind do the talking. The target makes their save based on the chart above. Your powers are limited to 60'. If they save, you are stunned for 3 rounds (in combat use) or 3 turns (non-combat use). If the target fails, the power works (per spell description or Mind Blast - which we would redo the chart)
Now, in accordance to the Alexis TestYourBullshit! rule, I have not tested these because I'm still noodling them. It's my (perhaps incorrect) suspicion that psionics are one of those campaign tilters, just like tossing a +3 Vorpal Sword into a pile 'o treasure, so I tend to tread lightly. Do these seem doable? To me they seem fun.