Your question:
Some decisions miners make are reflected in the blocks they solve. Is it possible to send those miners some extra BTC without knowing who they are? It seems obvious that you could just send some coin to the receiving address in the coinbase transaction, but usually, that address is for a large group of people. Since the extra BTC is intended for only the one who made the decision, sending to the coinbase address doesn't work.
A friend of mine said something to me like this: When you're in a mining pool, if yours is the machine that finds the solution to a block, then you get a much larger reward than everyone else. If this is true, then I would expect that the receiving address that gets the largest piece of a transaction that spends the total reward for a particular block is the address to which this "Thanks for mining the way you mine" reward would go. Can anyone confirm that this is the case, or identify cases in which it isn't true?
I would very much like a tool people could use to thank miners for particular decisions. I could write it myself or pay someone else to do it. I'll promise right now to send one bitcoin to anyone who provides such a tool.
If the miner was mining solo, then your suggestion of using the coinbase transaction works. However, as you correctly observe, the coinbase transaction of the majority of solved blocks either belongs to the pool on which the miner is mining, or in the case of Eligius and p2pool, every miner that will get a share of the block reward.
Your friend is incorrect. There are very few pools that reward the miner who finds the block. P2Pool does. The miner who finds the block gets 1/200 of the block reward in addition to any shares he currently has on the chain. Other pools, like mine, might give the miner who finds the block a bonus during certain promotions, but on my pool, that bonus comes from my own pocket.
Some pools do, however, provide statistics on who solved the block. For example, if you take a look at my pool, you can see who has recently solved blocks:
http://www.bravo-mining.com/index.php?page=statistics&action=pool. I know kano also provides this information (but you have to be registered and logged in), as does Eligius. Other pools you'd have to check. Now, just because you can see who solved the block, that doesn't necessarily give you the person's address to which you could send coins. I don't provide that correlation on my pool. Obviously I have access to that data as the pool operator, but it isn't public.