Rather than rolling a random value in a range, roll multiple times and sum or take the mean of the rolls to more heavily weight the middle of the range. (Assuming that’s a property that’s useful to you.) For example, the sum of 2d6 has a 1/6 chance to roll a 7, and only a 1/36 chance to roll a 2.
The effect increases as the number of rolls increases. The mean of the rolls is the Bates distribution. The sum of the rolls is the Irwin-Hall distribution.
Some games are built around this mechanic; Powered by the Apocalypse games often use 2d6 to favor the middle result.
A subtle instance of this occurs in Advance Wars. Most COs have a Luck stat of 10, where an extra 0–9% damage will be added to the preview damage of an attack.11 % damage is confusing here; damage is reported as a % of the target unit’s maximum. An attack that previews 45% has a chance of destroying a unit with 50% HP, since the attack will do 45–54% after the 0–9% luck is added. A few COs also have Bad Luck, which is rolled independently.
Sonja has a Luck of 10, but also a Bad Luck of 10. The mostly likely outcome outcome (1/10) is no change to damage. A 9% increase to damage occurs only 1/100 rolls.