I've used coins to add randomness to a base value. Flip two coins and add 1 to your stat for each head. This gave the limited amount of randomness that I was looking for. It was pbem and I wanted players to automatically be able to do easy stuff and know when they have no chance so that they could write their actions to cover more ground and make the game go faster.
There are various sorts of rock-paper-scissors which I've seen in board games and larps but I'm not sure about tabletop rpgs. This solution adds uncertainty but not randomness - dice add uncertainty by adding randomness. For example, both players in a fight pick a card to represent their action and puts it on the table. The outcome is a table lookup of the combo of choices. You can add tactics to this by selecting which cards you can play in chargen.
For my Cyberpunk contest entry I'll be trying a non-dice mechanics system based on betting and raising. The basic mechanics are as follows:
A character has three attributes (Physique, Acumen, and Anima), and there are some skills (hacking, modding, etc for cyberpunk). You get nine points to put in attributes, and perhaps half a dozen into skills (depends on the number of skill you have), which can have as many points in as you like each.
Each player starts with 20 chips at the start of an adventure. Each NPC has a certain number of chips depending on their power-level, as do obstacles (locked doors, etc) which can be described on the fly if you like. NPCs also have attributes, etc.
When an obstacle is encountered or a scene with an NPC plays out the player explains what they want their character to be doing ("I open the door"), and, for the most part, that's fine. When a circumstance comes up where an obstacle should impede them, however, the chips come into play.
The player says "I open the door", and the GM (throwing in one of the door's chips) says "It's locked". The player, wanting to get past, throws a chip of his own into the middle of the table and says "I hookup a dataline to the security panel and open the door". The GM at this point can continue to raise the action using the door's chips, or let the character pass. On obstacles, all the chips should really be used.
Attributes and skill add an extra dimension - every time you make a number of bids equal to your relevant attribute, the number of chips each bid costs doubles (attribute of 3 - after 3 bids for an action, it costs 2 chips, then after 6, four chips, etc), and skill points give you free raises.
I came up with a rather silly idea some week ago (perhaps it's been tested before): you take a pedometer, tell the other's what number you're aiming for (or use a calculated target number), and shake it real hard. Success is relative to how close you manage to get. Maybe best out of three?
Nifara, are the chips retained if the pc wins? Otherwise it's just one more way to spend resources. NPC's, like nonsentient obstacles, have no reason to try and retain chips. It's in their best interest to throw everything at the pc's. Without a bluffing element, or an uncertainty such as an unrevealed card, I don't see how a betting/raise mechanic would be beneficial.
Groffa, have you tried getting results from shaking a pedometer before? In my limited experience with them that doesn't seem to do much.