Duplicate questions are a drain on the trilogy community -- not necessarily a big drain, but a drain. Either you have to be a duplicate cop, or you find yourself answering a question only to have it closed as a duplicate.
I suggest these changes, the goal being to encourage people to search before posting:
Even before the question is closed, list the links to questions people have voted as duplicates for everyone (not just people with close rights) to see, including the questioner. Remove these if the question survives for a day (or change them to "related questions"). (Edit: I just went to post this as its own feature request, separate from the rest of this suggestion, because it has utility beyond the thrust of this idea. But someone already has [and quite some time ago]. If you have an opinion about this point separate from the rest, I recommend going and voting on / commenting on that question.)
If a question is closed as a duplicate, ding the questioner's reputation by a smallish, but real, amount; say -10. This can be avoided if the questioner sees people are voting to close it, looks at the questions they're pointing to, and deletes the question themselves. (Edit: Farseeker and others have pointed out that there should be a threshold so this doesn't hit newbies. I'm all in favor, that's a very good idea. It can be based either on rep, or on number of questions closed as duplicates. I prefer the latter -- Farseeker's suggestion was three, which seems reasonable.)
Obviously, if this isn't done already, reverse any rep gained from up-votes on the duplicate question. (Edit: Gnome points out that this is currently done, but only if the question ultimately gets deleted, which sets the bar very high indeed. I submit that it being closed is sufficient.)
Obviously, I did search before posting this! And reviewed the "Related Questions" list carefully, but to my surprise I didn't see this suggestion made.