Mona is a common English given name. It was fairly consistently a popular name in the UK and America until the 1970s, dropping out of the top 1000 most poplar names for girls in the USA in 1984. Its popularity peaked in the 1930s and again in the 50s (raking in the mid-200s in popularity), but these were not sharp peaks, and historically it had rarely been out of the top 500 most popular.
It has at least three distinct etymological origins. In Irish and English is comes from the Anglicized form of Muadhnait (pronounced MOO-nat), from the Irish muadh meaning 'noble', or 'good', with the feminine diminutive suffix -nait. It is generally translated as "little noble one".
In Scandinavian countries Mona is a common shortening of Monika or Monica, a mysterious name of unknown origins. It probably comes to us from a North African or Phoenician root, and was popularized in the west by the 4th century saint Monica of Hippo (mother of Saint Augustine). It has often been given the pseudoetymology of the Latin moneo, meaning 'advisor' or the Greek monos meaning 'one'.
In Arabic, the name Mona, Mouna, or Muna (منى) is the plural of munyah (منية), and means 'wishes', or 'desires'. It is a very popular name in many Islamic countries, but hard to find good data on; I have found unsubstantiated claims that it is the "fourth most common name for girls in the Arab world" and that it is the most common girls' name in Egypt.
There are many other roots for this name. In India, Mona may be used for boys or girls and again has multiple roots -- even within just the Hindi language. In the Western world, many Monas probably take their name from the Mona Lisa, in which Mona is not a given name but a polite title used to address a married woman, a shortening of ma donna. There are slews of other purported origins from various languages, but in many cases these are confused or made up; when in doubt, the intent behind a name is best requested from the person who gave the name.