Vault Hunter Relic is the only relic that improves rare item chance.
In Borderlands 2 you can find Basic and Legendary relics.
Basic Relics
Vitality: Increases health stats.
Stockpile: Increases your ammo storage for different types of ammo.
Protection: Improves shield capabilities and damage absorption.
Strength: Improves melee abilities.
Resistance: Limits damage you take from different types of weapons.
Tenacity: Intensifies Second Wind abilities.
Proficiency: Decreases the action skill cool down rate.
Elemental: Provides an increase to elemental weapon damage.
Aggression: Provides a boost to a particular weapon type.
Allegiance: Reserved manufacturers products improved.
Legendary Relics
Lucrative Opportunity
It causes shop timers to count down faster.
Deputy’s Badge
It increases “Fight For Your Life Time” by 10% for every player who has this relic equipped. Location: Showdown
Sheriff’s Badge
It increases “Fight For Your Life Time” by 15% for every player who has this relic on them. However, it is very rare and so more difficult to find. Location: Sheriff (Lynchwood)
Moxxi’s Endowment
It increases the amount of XP earned by killing enemies but is rare. Location: The Good The Bad And The Mordicai Mission, Moxxi’s Endowment
Vault Hunter
It enhances the probability of enemy dropping rare loot. It is also rare.
The Afterburner
It intensifies vehicle boost capabilities and it is rare too. Location: Positive Self-Image
The Vault Hunter's Relic only drops the chances of white item drops by 5% and then splits that 5% between the green and blue drops. so say if the odds of drops are:
No drop:40%, white:30%, green:15%, blue:10%, purple:5%, orange:Only on named & bosses (and often about 1% or less there) OR 3 borderlands symbols on a slot machine (about a 1 in 5,000 chance)
The Vault Hunters Relic would only make that:
No drop:40%, white:25%, green:18%, blue:12%, purple:5%, orange:Only on named & bosses (and often about 1% or less there) OR 3 borderlands symbols on a slot machine (about a 1 in 5,000 chance)
assuming those numbers were accurate (the orange is pretty close but the others I am mearly speculating) then even if it were modded to be a 30% better chance (thus eliminating white drops all together) it would still only be about: None:40%, White:0%, Green:38%, Blue:17%, Purple:5%, Orange:special.
No modding of %'s will make the Vault Hunters Relic really worth wearing once others become an option. Better to wear a relic that works to improve your strengths (such as an elemental boost on a siren or a brand-specific relic that helps boost your best weapon) or compensate for a weakness (+shield capacity or max health if you find yourself dying too easily).
Best Answer
Vault Hunter's Relic
It has a very minor effect on legendary drop rates, that you're probably better off using another relic. It only affects white (common) and non-white drop rates (and not legendary drop rates, specifically).
From this somethingawful.com forum post by user, gibbed:
True Vault Hunter Mode
Source: Borderlands wiki, "True Vault Hunter Mode" article