Two things come to my mind immediately: Artifacts in the original AD&D DMG like the Rod of Seven Parts, and an extension of artifacts that I used in a D&D 3.5 game. Hopefully you'll get some ideas that will work for your game.
The AD&D artifact route is fairly straightforward: as you get more pieces of the artifact together, you get access to more and more powerful powers. So as you get more parts of the Rod, or collect more teeth, or gather the saint's remains you get access to more powers. If each part has a different basic power, it may make sense to have different party members
"own" each item and pull them out to combine them as situations come up.
I don't think that's exactly what you're looking for, so I'll tell you about The Cubes.
In my game, the party stumbled onto a glass bracelet with a huge ruby in a treasure hoard. After some research, they discovered that its natural form is a glass cube with a smaller ruby cube inside it, hypercube-fashion. They discovered how to tune it to a specific person, and after some concentration and experimentation the wearer (he was a rogue, so it took him several levels of putting skill points in Concentration to do this) could get a direction on where the next piece lay.
Come to find out there are seven cubes, each dedicated to an ancient god. Each embodies an aspect of the appropriate god and provides a passive and active power to the wielder, but it must be in cube form in a hand to use the active power. So the Green Cube embodies Life and provides a continuous Death Ward aura, and it can be used to cast Heal once a day. The Red Cube embodies Fire so the wearer can heal when exposed to fire, and it can create a Meteor Swarm once a day. The Purple Cube embodies Magic so its wearer enjoys continual Arcane Sight, and it acts as a Rod of Spell Absorption up to 10 levels of spells.
Once the party started fitting the cubes together, they discovered that all the other cubes could adhere to the Purple Cube, and it would amplify or extend their powers using its stored spell levels. So a Purple/Green combo could increase the Death Ward aura into a Regeneration aura, or a Purple/Green/Red combo could call a green-tinged Flamestrike which would do double damage to undead but heal 1d8+5 HP to all living targets in the blast. The trick is, only the PC attuned to the cube can use it and control it. So multiple-cube effects requires several PCs concentrating to maintain contact with and control their particular cube.
Yes, we strayed from the rulebook a bit, but having the structure of World of Darkness Mage-esque keywords gave the players guidelines on what the cubes could do, and then their imagination took over. The life-powered Flamestrike was entirely my group's idea, but it fit within the realms of the cubes involved and the power level that the cubes could handle so it made sense to allow it. The purple cube can only absorb 10 spell levels, so a couple of 5th level effects would max out the effect unless the cube wielders wanted to tap into their active effects and "burn out" that power for 24 hours.
The spell levels of effects were pretty easy to figure in my head. There's usually a metamagic effect that's pretty close to what they wanted to make, or I thought of the effect as two spells cast at the same time. And yes, I granted bonus spell levels for really cool combo ideas.
These are really rough guidelines, and I never really codified the powers, but I took the basic inspiration from the Rod of Seven Parts, added effect keywords that I swiped from the Mage RPG, and fit the effects into the 3.5 system. I let my players flexibly define the combined powers, which got them really engaged in "I want to try this" playing as opposed to having them limited by their character sheets.
Your mileage will vary wildly, but hopefully you can get some ideas from my anecdotal ramblings.