Most gamers I know can name at least one influential heroes or villain's signature weapon: Lord of the Rings boasts wizard staves and unique swords, among other things; The Sword of Truth series contains magical items not found in any other cosmos; Excalibur from Arthurian legend and even Monty Python's Holy Hand Grenade are fantasy icons as well. Role-playing games can embody this type of inventiveness just as easily, with a few GM calls and some rules references. Whether your game calls for jimmy-rigged, custom commissioned, or personally crafted items, innovation makes for a more fun game. Don't forget about game balance in the mix.
On more than one occasion, groups I've played in used pints of oil and sacks to create large-scale molotov cocktails. Many old-school dungeon crawlers will tell you that a hammer is just a really heavy lockpick. There were even enchanted canteens which poured forth unending streams of water, serving as murder weapons inside the belly of a very large and powerful foe. All of these examples show the heart of creativity and using what's available. Using an item for an effective but unintended purpose is something players should be rewarded for and GMs should be mindful of. Especially work to thwart this if your group comes to rely on it for every encounter because then the game loses much of it's challenge and thought provoking nature.
Not every adventurer will be satisfied with a wand everyone else could own, or the same masterwork shield the king's guards are using. Most of those might get inventive and, for the right price, put up a merchant to make something specifically for them. Special enchantments are not unheard of and even equipment made from special materials can have very unusual properties. Take care that the merchant doesn't turn on the party and use their knowledge of the parties equipment against them, or even sabotage them during the making. A great craftsman could make a great living off of the adventuring party, though, so turncoats should be the exception rather than the rule.
The best avenue for uniqueness in equipment is when the party handcrafts everything themselves. While the wizard in town knows what each of his magical wands and staves are capable of, imagine the surprise when the party shows up with signature relics holding powers unknown to the antagonist (or anyone else, for that matter). This is the biggest reason I encourage my players to be blacksmiths, tanners and any other skilled, crafting trade -- there's just something awesome about a signature item that only the character has possessed throughout it's entire forged life.
While adding more items to the mix can seem frustrating for the GM, it should actually be seen as an opportunity. Remember that not every conceptual item actually works well in practice: while fireball can easily be cast from a wand, it would take a whole lot more work if it's even possible to get every blow from a sword to deliver that type of power. GMs should seek ways to say "yes, but" when players want to come up with new items, or use existing items in potentially disastrous ways. That explosion might be great but it's sure to attract a lot of attention -- some of which might be immune to fire. Sometimes players might wish they could relock a door they've previously broken into. Preserve immersion by allowing things to work to a point. No avenue should be inexhaustible. Happy inventing!