As Drmixable mentioned, it is very situational. If you're intending to solely be a kill lane, it might make sense to go Sapphire Crystal, ward, health pot, then tears.If you do this, you are gambling completely on your lane getting first blood and then compounding that advantage into laning dominance for the rest of the laning phase.
If you are wanting to be a more traditional support early game, then definitely do the standard Faerie Charm, wards, health pots mix.
Even if you do start with the mana crystal and go straight into tears, I think it still makes sense to move towards the more traditional support build after you pick up the tears. Philo stone, Heart of Gold, lots and lots of wards, oracles, etc. Build your gp10 items into their upgrades, prioritizing Shurelya's if your team is winning team fights and Locket if they are not.
And - as always - buy boots on your second trip back to base. For Blitzcrank, Boots of Mobility upgrade later makes a lot of sense for getting into position to yoink someone or knock them up to protect your carry. However, if the team is full of hard CC you will of course want the tenacity from merc treads instead.
Other good items include getting an early catalyst(which will build into Banshee's later), to sustain your lane harass and increase your passive shield proc if necessary. An aegis, of course, is always welcome from a support. Frozen Heart could be viable if the enemy team has more than two sources of straight auto attack damage, but generally won't be worth it.
Hopefully this wall of text gets you started on the path to thinking about how to apply your items to each situation. There is no single item guide that any hero should follow. Anything from who you are facing, how much gold you have, and how late it is in the game can vary your items and the order you build them in.
GL HF!
This answer was given on 05-02-2014. The current patch is 4.1. The answer will probably get less accurate and/or relevant in future patches.
I'm gonna have to make some assumptions to create a proper example here.
- I'm not taking in account any gold increases from masteries.
- I'm taking the first ten minutes of the game as 'laning phase', The
values may change if the laning phase is longer, but you really
can't know that up front so early decision making should not change.
- I'm not taking into account any jungle farming
- I'm assuming the 'level 1' version of the items, buying the upgrade
at some point will ofcourse change the data
Relic shield:
Minions start spawning from 1:30. So at ten minutes the 18th wave will spawn. This means that for the first ten minutes you have 17 waves to farm (the 18th will be underway at ten minutes). A cannon minion spawns in the wave every 3 waves and the Relic shield has a 60 second internal cooldown. So you can last hit the following:
wave 1 (melee minion) +23 +5
wave 3 (cannon minion) +40 +5
wave 5 (melee minion) +23 +5
wave 7 (cannon minion) +40 +5
wave 9 (melee minion) +23 +5
wave 11 (cannon minion) +40 +5
wave 13 (melee minion) +23 +5
wave 15 (cannon minion) +40 +5
wave 17 (melee minion) +23 +5
total 320 gold
Spellthief's edge:
The internal cooldown is 10 seconds per enemy. In a standard game you will be able to start hitting your opponents from about 2:00 when both teams go to their lane after helping their jungler. That means you can hit your opponents 48 times (both of them, as the cooldown is not shared between enemies hit) so 96 procs on the item. It gives you 4 gold per proc. Spellthief's edge also gives a passive 2 gold per 10 seconds. This is 120 gold for ten minutes.
total 504 gold
Ancient coin:
This one is easy, there are 17 waves (as said in the Relic shield part) to be farmed. Four of them have an extra minion. totaling 106 minions. The coin gives 2 gold. It also has the 2 gold per 10 seconds passive.
total 332 gold
Conclusion:
Spellthief's edge is the best item for pure gold gain, but it only grants gold to you. The Relic shield also gives gold to your adc (85). Ancient coin seems like the worst choice gold wise, but it's a lot more passive (and easy, 'safe pick' etc). Depending on when you buy the upgrades this coud change a lot. I think it's too complex to go into that in an answer here, if you really want to know about the upgrades as well, I suggest looking up a support guide that covers these items.
sources:
Relic shield wiki entry
Spellthief's edge wiki entry
Ancient coin wiki entry
Minions wiki entry
Best Answer
I'd Ancient Coin is the safest pick of all the 3 support items in general. Its gold income may not be the highest but the most reliable one (you just have to stand near minions) and its active works well with every champion/composition because you can use it to engage, disengage or catch people. Of course for Blitzcrank the bonus MS is especially nice since he relies on catching opponents.
Relic shield is more of a defensive item. You get a decent amount of gold, a shield and what most people forget: sustain. The sustain is more important than ever before, since Dorans got nerfed in early stages. Of course relic shield is preferred by Meele champions, since you get the below-200-hp execute bonus. On Blitzcrank I'd say you should buy it if you want to go more tanky, if you're getting poked sometimes and if you are vs another All-In botlane (Leona, Alistar, Braum, Thresh sometimes). The Active and the heal are both really good but you still have to lasthit minions to get the bonus gold.
The Spellthiefs Edge... well what should I say. It's a great item but if you're not in Troll mood, don't even think about buying it on a meele character. While its active may synergize with Blitzcrank you are better off with buying more tanky or more utility.
So basically When do you want to buy which item?
Ancient coin
Relic shield
Spellthiefs Edge
So I'd say you can really buy the coin all the time and against/with every team. You can't really do anything wrong there. Maybe this is the reason why a lot of people consider it to be the best Item on Blitz. However in some situations a relic shield will do just as fine if not better.