I suggested a tag synonym several days ago which was accepted, but I haven't yet been awarded the Synonymizer badge. When will this badge appear in my profile? I thought it would have appeared by now.
The badge is only awarded through creating a tag synonym through the tag system (which requires 2500 rep and 5 votes in the tag), not recommending one in meta.
To expand upon Shane's question, the official tag synonym system is not, "Post to meta, have a mod do it." It actually involves having enough rep to suggest a synonym, and having enough users upvote the synonym that it becomes an official synonym.
This can be short-circuited by suggesting a synonym via the official system, and then a mod later approves the synonym.
Because of the reputation requirements for synonyms, I believe this method is the only way this badge is being earned these days. The synonym system was written for SO, and even though we're the #3 site for traffic, it doesn't work for us as written; the mods end up doing all of the synonym creation.