Here's my logic: when you have control of the game, it doesn't matter what your finisher is, so you should pick something that helps you out as much as possible when you don't have control. (I originally read this in Randy Buehler's description of his 1998 Worlds deck, in which he used Rainbow Efreet.) This holds true for essentially all of the common control finishers in Standard today:
- Consecrated Sphinx lets you draw a ton of cards. Even using its ability two or three times can give you enough card advantage to match your opponent spell-for-threat.
- Wurmcoil Engine gains you lots of life and also can trade for up to 3 of your opponent's creatures or removal spells, which is a huge setback for an aggressive deck. This is why it's a popular finisher against aggro decks, but not so much against other control decks or combo decks.
- Sun Titan lets you get an extra use out of one or more of your spells, often something like Oblivion Ring or Ghost Quarter, which can help you regain control when facing down troublesome permanents. It plays a similar role to Consecrated Sphinx
- Grave Titan is basically a horde-in-a-box (actually in a card): getting three creatures out of one card, especially given that one is giant and has deathtouch, can shut your opponent's entire attacking force down. Plus, if they don't deal with it immediately, you get two more creatures every turn.
- Going back a bit, Celestial Colonnade helps you when you don't have control in a different way: it can't be countered, and it requires instant-speed removal to get rid of. This is more of an anti-control finisher, since when you're playing another control deck, not having control of the board generally means that you're outmatched with counterspells or removal spells. The Colonnade lets you put pressure on your opponent without having to fight through a ton of counterspells. (Same goes for any other "manland.")
- Morphling was a good anti-control finisher for a similar reason to the manlands: the ability to give it shroud and pump its toughness made it very difficult to kill.
- Sword of Feast and Famine was the namesake finisher of the Cawblade deck. Its power came from the fact that it allowed you to get double use out of your mana, once for advancing your own board and once for countering your opponent's spells. So even if you were somewhat behind in the game state, once you started getting Sword hits in it became fairly easy to recover. (To be fair, Cawblade wasn't a true control deck, but it did fall into that role sometimes.)
- Planeswalkers generally tend to have very powerful abilities that can go a long way toward helping you gain control of the game. This is why Big Jace was so good: it could bounce a creature, draw you cards, or shut your opponent out of good draws. It was like the control player's Swiss army knife. Gideon Jura is a popular choice these days because it can stop your opponent from attacking you, saving you some life, and it acts as a repeatable removal spell as well, so it's difficult for any creature-based deck to play around it. And of course, with both of those planeswalkers, once you've used their removal capabilities to their full extent, they give you a direct win condition as well.
You'll notice that all these cards have big, game-changing effects: 6-power creatures, 4-power flyers, removing your opponent's entire library from the game, etc. This means that if they're not dealt with, they let you win very quickly. There is something to be said for this strategy, because it minimizes the amount of time that you need to maintain control of the game, but in a proper control deck it's not necessary to have a big flashy finisher. Something like Fiend Hunter, for example, would not be too bad because it exiles one of your opponent's creatures and then can sit around and block. Back in the days of the original Caw-go deck, even Squadron Hawk filled the finisher role well because you could play one to attack and it came with 3 buddies who could either block or add to the pressure on your opponent. However, Wizards is making it rather difficult to create a true control deck these days, mostly because counterspells are just not as efficient as they used to be. So I wouldn't actually suggest using something like Fiend Hunter as your only finisher.
When choosing a finisher, the cards I would stay from are those that don't do anything to help you regain control of the game when you are losing it. As a concrete example, you could imagine "losing it" to mean that you're at 4 life and facing down three 2-power creatures; or perhaps you're at 7 life and your opponent will be able to cast Devil's Play to burn you out in two turns. Kindercatch would be a great example of a bad finisher because it just doesn't do anything to help you get out of either situation. Sure, it can block one of your opponent's creatures, but you need something to deal with multiple creatures and non-creature threats.
I would actually say something like Progenitus is only a mediocre finisher, because the only sense in which it deals with anything is by killing your opponent before they get the chance to cast that lethal Devil's Play or whatever. Admittedly it is very effective at that, but you have to consider that a giant creature like that is of absolutely no use in the early game, before you've accumulated the 10 mana required to cast it, which is when you typically need the secondary effects of your finishers most.
EDIT: See this video, specifically game 1 of match 3, for a great example of Snapcaster Mage being used as a finisher.