In my Doctors of Lisbon home game, I built a complicated but fair ability score generation procedure. Maybe it has the sorts of advantages you are looking for in your home games?
Advantages:
- Everyone at the table gets a character with a total number of ability score points that matches everyone else. This means that the characters are roughly equal in power, since power increases and decreases linearly with higher and lower ability scores.
- The stats generated are mostly random. It is my hypothesis that stats generated semi-randomly make characters that interest 'concept-first' players and also 'numbers-first' players when generating characters. Players that like to start with a concept and roll dice don't like it when their character concept isn't possible, and players that like to start with numbers first don't like it when the numbers are too flexibly arranged (I think of this as a choice-paralysis problem).
- You get to roll your dice instead of just assigning numbers.
- Every character has slightly inoptimal stats, but not painfully inoptimal.
Disadvantages:
- It is complicated compared to other methods. This is a pain. If I thought about the math some more I might be able to make it less complicated while still accomplishing the same goal.
- You only get to roll 1d6 at a time.
- All characters are roughly equal in power. I'm willing to take this disadvantage in my game.
- You don't get stunningly inept characters. This will be a problem if you are running a game set in a crapsack world. I'm willing to take this disadvantage in this specific campaign.
Custom Stat Generation Rules
This weird stat generation trick gets everyone fair stats, but it has more randomness than point-buy or fixed array. My apologies for the complexity. Your average stat is still a 13, same as the traditional 4d6 drop lowest.
Roll on this table 12 times and count the number of times you land on each value:
If you land on one of these values more than 5 times while rolling your twelve times, reroll that time until you get a different value - these rerolls don’t count against the 12 original rolls.
After these rolls, you should have a little table of numbers showing 0-5 for each stat like this:
Str: 2
Dex: 2
Con: 2
Int: 0
Wis: 5
Cha: 1
Subtract 1 from each stat - it should look like this:
Str: 1
Dex: 1
Con: 1
Int: -1
Wis: 4
Cha: 0
Now the tricky part - convert these to the even stats that represent these as if they were modifiers according to normal D&D stats like this:
Str: 12
Dex: 12
Con: 12
Int: 8
Wis: 18
Cha: 10
These are the stats for your character before racial adjustments.
OPTIONAL: You may choose any 1 set of 2 stats and swap them. Here I swap Dex and Wis:
Str: 12
Dex: 18
Con: 12
Int: 8
Wis: 12
Cha: 10
OPTIONAL: If your Con stat is 8, you may choose to subtract 2 from either your highest stat or your second highest stat and add it to Con to increase it to 10.
Last Step! Roll on this table until you get 3 unique stats, then add +1 to each of your stats that correspond to the table. Reroll any values that would push you over 18 in any one stat.
My example stats are:
Str: 12
Dex: 18
Con: 12
Int: 8
Wis: 12
Cha: 10
I roll a 1, a 4, and a 4, so I roll again and get a 6. I increase my stats like this:
Str: 13
Dex: 18
Con: 12
Int: 9
Wis: 12
Cha: 11
