d20 Point Buy – Wild Card Option
Using point buy to assign ability scores is more balanced than rolling, so you avoid having some players get lucky and while others are penalized. It also gives the players more control to create exactly the character they want. The problem is that the players focus too much on the scores that are important to them. The hyper-specialized characters that come out are too perfect, with not a single point wasted on an ability that doesn't forward the concept.
A compromise is to use point buy, but with a small element of balanced randomness. The player rolls for one ability score that is not essential to the character concept but gets to assign the rest.
1) Player has their character idea, but has not assigned any ability scores. Player picks 2 ability scores that they definitely want to keep control over. These are usually abilities critical to the character concept.
2) Roll d4, the result indicating one of the four remaining ability scores in order.
3) Roll another d4-2, which is then the bonus for that ability score (resulting in a -1 to +2, or 8 to 14 assuming we're sticking with even scores — you can adjust to include odd scores if you want). That ability score is fixed and cannot be changed by the player during point buy.
4) Determine the cost to buy that ability score and subtract that from the points you start with. The player can spend the remaining points on other ability scores however they want.
Since the player is paying for the random ability score normally, all characters will still come out balanced by cost. The one random score just throws in a little unpredictability, increasing the chance that a character will have a quirky or idiosyncratic score.
Morrik the bandit decides that Str and Dex are vital to his character concept, so those are the ability scores he wants to keep control over. That leaves Con, Int, Wis and Cha in order. He rolls a d4 and gets a 2, so Int will be his random ability score. He rolls again and gets a 4, which -2 is 2, so he has a +2 bonus (Int 14). The DM is giving all characters 26 points to spend, and since his Int costs 6 points that leaves him with 20 to spend on his other abilities.
A standard point buy has abilities start at 8, and costs 1 point to raise an ability score 1 up through 14. Raising to 15 or 16 costs 2 each, then raising to 17 or 18 costs 3 each.