There's an idea that appeals to me which uses the best part of dice but which extends it outward, beyond the limitations of dice.
Dice are limited in many ways, in order to get a rollable dice you need the sides to be regular and the chance of any particular face coming up to be even. You're limited, then, to the typical, regular polyhedra and their ranges. You can add dice together, to get a nice bell-curve of probability from them, and you can even create cylindrical dice with a wider variety of sides (as the Romans used to) but you're still somewhat constricted by the dice and by adding, taking away and less commonly multiplying and dividing,
Sometimes you can pit one dice against another, which can be a good way of modelling low-skill and high luck, versus high-skill and low luck – depending on the vagaries of the roll.
There's no reason – necessarily, that we be bound by this any longer however. We can create 'conceptual dice' via the means of random number generators and give them any values we like, no longer bound by the physical world and the necessity of a regular polyhedron.
Instead of 1d12+2 damage, we could roll a D14 – allowing for the possibility of truly glancing blows. We could pit a Stat+Skill of 1d11 versus a Stat+Skill of 1d17. If you want that bell-curve distribution you could get it by adding two results together and they needn't even match.
'Ditching the dice' opens up a world of possibilities, and going for random number generation retains the utility of dice, without their limitations and without the faffing around and compromises that non-dice systems tend to fall into.
Of course, getting players to give up their totemic and beloved dice won't be easy, even though many are already using phones and tablets with dice-apps already.
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