In Japan the character is known as "Dr. Right". The spelling "Light" came about because there is no distinction between the L-sound and the R-sound in the Japanese language. This caused some confusion in early translations: originally, in America, the character was renamed "Wright" because Capcom of America thought that the Japanese had tried to imitate American names unsuccessfully (this is also why Wily is often confused as "Wiley"). Dr. Light has been called by his Japanese name only once in the English-version releases, in Mega Man 3 (NES, 1990).
Further complicating matters, in the Mega Man Battle Networkalternate universe series, Dr. Light's character counterpart is named "Tadashi Hikari" (光 正 Hikari Tadashi). Hikari (光) means "light", but Tadashii (正しい) means "right" or "correct". Because of this, it seems Capcom's official stance on the matter is that both "Right" and "Light" are correct.
It is common in the American fandom to attribute the middle name "Xavier" to Dr. Light, but this name is not found in any Japanese or American official sources.