This is just a small note to put on record a pretty far-fetched theory. As I pointed out on this board before, I do not know the origin of riyA, and I have no dictionary or usage which shows riyA as a singer in Sanskrit. But, Sanskrit is a big language, and I have over time figured out one possible way riyA can have something to do with singing.
There is a root R (ऋ) which means to move up or be straight. This is the root of words like Arya (from someone with straight or high conduct) and Rju (straight as in a simple path) etc. But, an obscure use of the word is to raise ones voice in singing vedic chants in praise of the gods. This usage could give rise to riyA meaning chanting or singing. If I tried a bit more, I might be able to get singer, but as I said this is already far fetched, and I only wanted to note it since I haven't seen this written down elsewhere.
I am, however, still inclined to believe that this is a foreign name borrowed into India. The earliest Riya I personally know of is probably born around the middle of the 20th century, and spells her name as RYA (ঋয়া) in Bengali, further confirming the theory that it is not Indian (The sound R is rather uncommon in newly coined words), or it is related to Sanskrit R. In north India, R often merges into ri in pronouciation (elsewhere, it often merges into ru).