You are right that Indu spreang to mind when I saw this thread. I doubt that the name contains the element deva. Indian phonology, as far as I understand it, tends not to omit a pair of identical consonants separated by a vowel, so indudeva, even if it ever was used, is unlikely to have become indeva. It is far more common to see the u to v change on addition of vowel suffix: in fact since Indu is masculine, if I wanted to make a feminine today (not grammatically, just what sounds right), I probably would have come up with indvi or indavi. Similarly, in Sanskrit indavah is used as a plural of moon. (indu probably literally meant a drop, from ind, to drop: the moon was conceived as a drop of the invigorating drug soma consumed by ancient Indians in religious ceremonies).
Of course, it could instead be shortening of words like induvadana (moon-faced, from vad, to speak) or may not even be of Sanskrit origin. India is a vast country, and its languages merge words from Indo-european, Dravidian, Austrasiatic, Sino-tibetan, and Semitic language families (in addition to the occasional isolated borrowing). More information than merely `Indian' would help in this case.
Incidentally, I do not find it surprising to call a daughter "Night Rain".