There are so many bad remakes that it can be forgotten that there can be some good ones. What are your favorites? The Coen Brothers don't like calling it a remake, but their True Grit puts the original to shame. Let the Right One In was very good but the American remake Let Me In succeeds by getting rid of a couple bad scenes.
3:10 to Yuma and Insomnia were both good. Ocean's Eleven and Assault on Precinct 13 come to mind too. I'd say The Departed and The Birdcage but I haven't seen the originals to compare yet Are we counting things that started as book adaptations? Little Women, Emma, Dune, All Quiet on the Western Front.
That is always tricky. That is the issue with True Grit; the Coens insist they were adapting the book and not remaking the 1968 movie. Obviously if a Hamlet movie came out this year we wouldn't call it a remake of earlier films, but some examples owe a lot of their creative choices to earlier films.
If we're talking games, Lunar and Lunar 2 on the PSX are the definitive versions in my mind and improve upon the Sega CD games in every reasonable way.
The Shadow of the Colossus remake kept what was beautiful about the original game but managed to make it accessible for modern players which is all you can want in a faithful remake. Something like the Final Fantasy VII Remake is a different story; my familiarity with the original game made the padding all too obvious.
A few random ones off the top of my head; Shin Godzilla (Hideaki Anno, 2016): A Kaiju film that also focuses on the bureaucracy of the destruction of the city, and also gives the monster a fresh parallel with the recent Fukushima meltdown. There's a Shin Ultraman that was released last year, that I'd like to see. Nosferatu the Vampyre (Werner Herzog, 1979): Herzog taking the original Murnau film, and playing it fairly straight, with genuinely unnerving moments, and a great Dracula turn from Klaus Kinski. Hex (Kuei Chih-Hung, 1980): A weird, Hong Kong Black Magic take on the terrifying French film Les Diaboliques from 1955, that has some great set design and full of tension. Spawned a couple of knockabout sequels, but this is a intriguing watch. A Woman, a Gun and a Noodle Shop (Zhang Yimou, 2009): Zhang Yimou remaking the Coen's Blood Simple, and turning it into a farcical, screwball comedy, with elements of wuxia. It's not as good as Blood Simple, but it feels like a very fun reinterpretation. The Mummy (Stephen Sommers, 1999): Cheating a bit here, as I haven't seen the 1930s original, but The Mummy is basically a perfect adventure blockbuster, that I might prefer to Indiana Jones on some days. Ghost in the Shell (Rupert Sanders, 2017): While I fully understand the controversy about casting Scarlett Johansson in the lead role, I think the controversy of that overshadowed the fact that this was actually a pretty good blockbuster sci-fi film, and Johansson is great (even if she probably shouldn't have been cast in the first place).
Movies: The Thing Evil Dead Dawn of the Dead Video Games: Resident Evil Resident Evil 2 Expecting Resident Evil 4 and Silent Hill 2 remakes to be great. Need to play the Dead Space remake that just came out.