Thankfully this one has a short, simple answer. The English name for the country was borrowed from French in the 19th century, and in French it was (and still is) spelled "Roumanie". As a result, the original English spelling became "Rumania". After World War II, "Rumania" was generally replaced by the official native spelling, "Romania" (România in Romanian), but the archaic form was still used occasionally, even by specialists, for many years afterward (as demonstrated in the title of Keith Hitchins' 1994 book Rumania, 1866-1947). It's also worth noting that in many Germanic and Slavic languages, it's still spelled with a "u", e.g. German (Rumänien) and Polish (Rumunia).

Prior to the standardization of Romanian spelling in the early 20th century, the "u" form was sometimes used as an endonym as well (rumân vs. the now-standard român). The native form of the country's name underwent spelling changes as recently as the early 1960s, when the Communist regime went back and forth on the use of the letters â and î, which represent the same sound in Romanian (this sound doesn't exist in English but is close to the "e" in "roses"). In 1953, the Communist government replaced â with î in all cases, briefly changing the native spelling of the country's name to Romînia, but it was reverted to the original spelling in 1964. Since the end of the Communist era, î is only used in word-initial positions and â is used everywhere else (cf. înalt ("tall") vs. cuvânt ("word")). This was probably more than you cared to know about the Romanian alphabet, but I figured I'd add it for the sake of being "in depth".