There are some items that should exist but don't.  Names don't matter.  With no further ado:

Basic Items: Completing the List

  • Quicktime Charm: 180g.  Provides 180g worth of CDR.  Builds into Kindlegem, Brutalizer, Codex, Stinger, and whatever else gets CDR out of nowhere.
  • Vampiric Charm: 400g; 6% Life Steal (or 180g; 2% Life Steal).  Because sometimes you want that BEFORE your +10 AD.  Numbers are low because (1) compare the investment to LS quints vs. AD runes - 400g for 10% is too much; and (2) generally, items become more gold efficient when they're combined, so building a Vamp Scepter would follow suit.
  • Piercing Dagger: 300g.  Percentile armor pen.

Gameplay Holes: Serving a Purpose

  • Boots of Force:  Advanced.  Percentile armor pen.
    • Geard toward AD Casters, whom current item choices frequently disappoint. Percentile pen means that they're not broken to hell and back early game.
  • Someone's Rapier:  Legendary.  Expensive.  Grants +an assload (80?) of AD and maybe spell vamp (because it's expensive and ADCs don't want it).  Unique toggle: Your Attack Damage is increased X%; your Attack Speed is decreased X%.
    • The fact that Riven and Talon and Zed stack bloodthirsters is indicative of a hole in AD Caster itemization.
    • The toggle stops it from being an amazing item for ADCs; ADCs are fine as-is.
  • The Gapcloser: Legendary. Expensive (3600-3800?). Fucktons of AD, maybe some armor pen for kicks (gives Brutalizer another build path?). Active: Melee only. For 4 seconds gain +60% movement speed while moving toward foes that decays over the duration. 60-second cooldown.
  • Cleansing Potion:  Consumable.  Relatively expensive.  Removes CC.
    • In those matches where you need to build a QSS, you can't reasonably be expected to afford it until rather late in the game without being put substantially behind (it costs as much as a BF).  This helps.  And no, Cleanse isn't always enough.
  • Sentry Ward:  25g.  Places a ward for up to 90 seconds that dies in 2 hits, granting 20g to its killer.  This ward is not stealthed.
    • Single-use wards for peace of mind.  Saves 50g for level-one scouting and counterjungling, which makes a difference, but does a bit to make up for the gold that it's denying junglers in kills and assists.
  • Diamond Signtstone:  7 charges; 4 wards at a time.  Wards placed reveal stealthed units for 2 seconds when placed (15-second cooldown on reveal).
    • This is one of the things that would go a long way toward making playing support not suck.  Also combats global ward saturation.  Warding is no longer just "a thing I have to do"; it becomes a mini-objective, with (most of the time) two champs necessary to take down a ward.
  • Predator's Machete:  Advanced.  Gains stacks on killing a neutral monster; loses stacks on killing a minion.  Consumes stacks to grant gold on champion kill or assist.
    • Rewards successful ganks. Stacks should have diminishing returns, like Thresh souls, to encourage ganking frequently, rather than "omg need more stacks". Also does a bit to address the "junglers don't have gold" issue. Ideally, should build into useful items.
  • Thief's Gloves:  Advanced?  Grants gold on dealing damage to a champion; x-second cooldown.  Goes on cooldown on using a basic attack.
    • Another support item.  Ideally, it can build into some nifty things. The gold gain on hit would have to be significant or, alternatively, the gloves would have to offer reasonable stats on their own.
  • SightSword/SightStaff/SightShield: Legendary.  Builds out of Sightstone; grants stats that allow supports to do damage and feel powerful.
    • Can't give enough stats that they become commonplace; when this is replacing BT or Hourglass, there's a problem.

Just Concepts: Finding a Use

  • Crystalline Purse.  Advanced item.  Builds out of Flask.  Unique active: consumes a charge or, if no charges remain, 35g of your current gold to grant reasonable health/mana/energy regeneration (two pots worth?).
  • Consuming Blade/Staff/Shield/Shroud/Whatever.  Legendary, expensive.  Sacrifice your current gold; gain x% of that as AD/AP/AR/MR/Whatever.
    • For those 70-minute games.
    • Also useful lategame item for supports, if vision is fixed such that it doesn't cost all the money.