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?
- Pretty safe all the time
- If your team is rather movementspeed based
- If you need a stronger engage/disengage
- If you like standing in a bush and gaining gold for it
- If you/your adc are getting poked a bit
- If you are against an All-In botlane
- If you want to build more tanky
- If you are trolling
- Full AP Slow Burst Pewpew Blitzcrank I guess?
- Seriously don't do It on BC
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.