I have theory crafted a support build that requires 8300 gold to be fully effective. I want to know what support item gets me there the fastest. I do not care about the stats the support items may or may not give me, all I care about is getting that 8300 as fast as possible. So, what support item is best for the job?
To answer your thread question:
Personally besides using 
and I generally like 
just because if your team is chasing someone and you press Frost Queen generally they have no escape due to the vision of them + slow. This can result in multiple kills for your team and can benefit them greatly :D
Assuming the options are the talisman line versus relic verus fqc, I'm pretty sure relic is the best potential gold assuming you get to last hit every cannon minion.
That said, what's best in practice varies heavily based on support. If you're a ranged champ who can consistently poke whenever you have a charge you can make money fairly consistently with the fqc line. If you're a squishy utility, hanging back with the coin line might be the more reliable option.
That said, gold from the support items isn't necessarily the best route to getting gold. Not dieing/getting zoned counts for a lot, as does getting kill/assist gold, which is why sightstone is getting mentioned a lot. (There are even those out there who all in on the getting kills thing and just buy straight AP items [or AD I guess if an ad 'support'] even as a support in an effort to get kills. I don't like those 'supports', but if your lane opponents are bad, then it can certainly lead to a profitable snowball.)
If the LoL wiki is correct 
8.9 Gold per 10 seconds to the user, while also sharing that same gold with the lane partner.
Between 20 and 35 minutes, it provides 10 Gold per 10 to the user, while also sharing that same gold with the lane partner.
After 35 minutes, it provides 13.3 Gold per 10 to the user, while also sharing that same gold with the lane partner.
So gives between **11.9 and 15.3 gold per 10 **(though 15.3 is only after 35 min)
If you are fighting/hitting enemy towers every 30 seconds 
will give up to 17 Gold per 10 (15 gold per stack and you gain 1 stack every 10 seconds + 2 gold per 10)
gives about 13 gold per 10 though roaming will hurt it more then it hurts the others as they can use all their stacks before roaming and thus not lose as much
As support item I infer you are speaking gold first ( sup rarely farm).
I have tried 
.
- Upgrade to or 
. Gold coin is good if you are ALWAYS in lane with adc to harvest that gold. It's not possible to be in lane always and as a sup you might need to roam between lanes/jg. goodbye free gold. Final items are good but doesn't give any AP or AD.. so in short your damage is equal to minions (yup that's true)
- Upgrades to - Pretty good when chasing, running away or finding hidden champs. or - decent AP, CDR, Mana regen (not bad)
- Upgrades to - Good to tanky sup for health with no CDR or 
- Better option with CDR and health. These items are good with melee support due to health and gold gen ability.
I have played all three and found to be least useful in early-mid game. With i get decent gold and much needed AP to poke or do noticeable damage, not a lot but enough to kill any champ, with <30% health, with 2-3 ability combos. Not to forget has helped me >80% times. And Mid lane APs used to abuse this before RIOT nerfed it a bit.
If I'm going Tanky sup, then health is first priority so I go .
I personally will stay away from any of the Eye items, unless all my runes+mastries give me 40 AP ( which is possible at expense of MR/Armor/CDR).
Almost everyone expects support to have 
, be always present in teamfights ( that means farming in other lane/split pushing is not an option).
