The Humble d6

Rolling d6s for damage and hit-points
By design or blunder the white box delivers a fun game. The alternative combat system using a d20 to hit and a d6 for damage combined with hit points that generally increase with each level gained seems genius design to me and is a big part of why the white box is my favorite edition. Basing both the hit die and weapon damage on a d6 allows for a degree of predictability as well as randomness. I think both are necessary for maximum fun.
Gamers like to roll dice. The chance of rolling well is part of the thrill of our hobby. Risk, however, must accompany any game mechanic involving chance and there is the ever present risk of rolling a "poor" result every time we grasp the die. The unusual shaped d20 is a random number generator with twenty possible outcomes. Basically it works like percentages with a 5% increment between each possible result. The alternative combat mechanic (which would become standard in later editions) calls for the PC or monster to roll a certain number or higher on the d20. The target number is found on a chart and is determined by the class and level of the PC or hit dice of the monster and the Armor Class (AC) of the target.
If a hit is achieved (and celebration ensues) damage is then determined by rolling a d6 and deducting that many hit points from the target's total. PCs and monsters start with 1-6 hit points (plus or minus) so a single successful hit has the potential to take-out a low level PC or monster.  Seems "realistic". It isn't hard for me to imagine that a single wound from a sword, mace, spear or whatever could drop me and therefore seems believable as a game mechanic.
Now we come to the genius part. Higher level PCs and monsters have higher numbers of hit dice and therefore higher hit point totals. They may also become better able to hit an opponent as the target number on the d20 To-Hit roll goes down as PC levels and monster hit dice increase. The chance to successfully hit one's target increases 5% with each number on the d20. Thus a target number of 16 or higher (25% chance of success) represents a 5% better chance of hitting than a target number of 17 or higher (20% chance of success). This seems a stroke of genius to me because players can judge about how many hits they can take and roughly what the chances of success they have against a monster or many monsters, but it takes some thinking and it involves some degree of chance. With 1-6 points of damage per successful hit, the player who knows his/her PC has 10 remaining hit points can survive one hit, but might go down if they take two hits. Some games have made this even more predictable by assigning each weapon a damage value that never changes. If the player also knows they have a 25% chance of being damaged each time the baddies make an attack, they can make some reasonable choices regarding tactics (fight or flight?) and use of resources (whether to cast that high-level spell) to influence the outcome in their favor.
As editions add more variables to the numbers, variable hit-point and weapon damage dice (d4, d6, d8, etc), more adds to damage, feats and combinations thereof, etc. it has become harder for this player/referee to judge how many more hits a PC can take or exactly who's winning or losing the fight. Unpredictability/randomness can be fun, but when it comes to PC life and death, as a player and as a referee, I have found a simple satisfaction in being better able to grasp the numbers in my head and make decisions based on likelihood of surviving another x-number of rounds. And I like the idea that any weapon can possibly take out any 1st level with one lucky hit.