I love it as a middle name! For a first name, River Mai (pronounced May) is kinda cute. This is funny because we are thinking of naming our little girl Mya or Maia Oceana. I posted about oceana as a first name a few weeks ago, but we are thinking it is better as a middle name. I'm just worried about people pronouncing Maia correctly (I want it to be said MY-ah. So we might go with the Mya spelling.

I know an older lady and her name is Rivers.

I think River is a unique name for a girl. Why not!
Maybe you could give in and have her middle name Mya, or save that one for another child.
River Nicole, River Breanne, River leann, River Shay,many names could go. Just go thru a book, and try saying River with a lot of names and see what rolls the best.