To start off, I do not know. I am merely posting an uninformed speculation under the assumption that it is a name from the South Indian
Christian community: a very old (some say back to 3rd century AD, certainly not later by more than a few centuries after that)
Christian community called Syrian Christians. They have maintained (or absorbed, I do not know which) a large amount of social practices from the local religion (usually called Hinduism). The local languages are of the Dravidian group with large borrowings, especially of names and religious terminology, from Sanskrit and its Indoaryan daughters.
There is a Sanskrit verbal root R meaning to go towards or rise (cognate Indoeuropean roots mean things like to row or plough), whose causative stem arp(i)- is far more common in later languages and means (originally) to insert, entrust, deliver, or offer. Its past participle is arpita which, in later language, is better translated as dedicated.
In many South Indian languages, the -i- sound becomes -u- (as in put) in this context, and th is the preferred South Indian orthography of this dental unaspirated consonant (to distinguish from the retroflex variety: North Indians translitaeration likes to maintain the discrimination along the aspirated/unaspirated axis instead). Thus, we get the element Arputha.
Mary is the form of the Hebrew
Miriam which is common in
India (though I do not know whether that is true of the Syrian
Christian community historically). I also do not know much about Semitic or Egyptian, so look it up on the database, it may originate in Egyptian to mean something like loved or beloved.
And this whole long post might turn out to be completely wrong!