I can't be certain what the first remake was -- Japan's cinematic history is every bit as long as America's (sources trace the first Japanese-made film back to 1897) -- but the first really notable remake, the first one we still really talk about, was pretty far from the modern J-horror hits you're referring to. When people talk about Japanese cinema's influence on Hollywood, they talk first about "The Magnificent Seven."
Released in 1960, that film is still held as a landmark in the western genre and now considered one of the greatest American films of all time -- it was selected for preservation by the Library of Congress in 2013. (It was also remade again in 2016.)
Despite all these Americana-type accolades, the film was actually based on a Japanese film called "Seven Samurai" (1954). That film is probably a more iconic piece of Japanese film history and is considered one of the greatest films by that country's best-known director, Akira Kurosawa.
This highlights a fact worth noting: Hollywood's current tendency to mine Japan's horror genre for new ideas is just a repeat of the previous trend of mining the country's samurai genre. Some other American classics are based on other samurai films -- mostly notably "Star Wars" (1977), which was based (a bit more loosely) on "The Hidden Fortress" (1958), another Kurosawa flick.
So, naming the first remake is a challenge, but it might also be less interesting than finding the next remake, as that could be the start of yet another fruitful trend.
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