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True Grit would be on my list.
And Casino Royale.
I definitely agree with True Grit, (mostly for Hailee Steinfeld's performance over Kim Darby), but I'm not sure that I'd consider Casino Royale a remake given that the stories are different and the first one was a complete satire. But if we're going to consider remakes where one movie is serious and one is a comedy, then #1 on the list has to be Airplane! over Zero Hour!
Other remakes on my list that were as good or better than the original would include Invasion of the Body Snatchers, The Thing, His Girl Friday, Little Shop of Horrors, and Scarface.