The most unloved dice in the game are the d4 and the d12. What would you use them for so that they can see more action in the game if you were designing your own old school D&D clone?
I would use d4's where I wanted low variation in the mechanic and an average result of 2.5 per die. I would use d12's where I wanted high variation and an average result of 6.5 of per die.
So, for example, if I was starting from scratch, a spell that produced a cloud of fire would probably have d4 denominated damage, representing everything being roughly evenly cooked. A spell that on the other hand flung shards of metal about would probably have d12 denominated damage, representing random variation in the size and density of the shards that hit you. We could keep these spells roughly balanced in terms of damage output.
For example, the firecloud spell of a certain wizard might do 12d4 damage, for an average of 30. The schrapnel blast spell of the same wizard might do 4d12 damage, for an average of 26 or perhaps 6d12 damage for an average of 39 (with the choice being based on how relatively easy it was to protect against fire and physical damage).
In 1e where dice often lived in isolation (no modifiers), I often used different combinations of dice purely to get a slowly advancing increment of improvement, as in:
1d10 < 1d6+1d4 < 1d12 < 2d6 < 3d4 < 2d6+1
to represent increasingly dangerous claw attacks (for example).
And sometimes were the modifiers are fixed, you just use increasing die size to represent increasing magnitude. 3e does this with HD and weapons, for example.