Was Pharoh's daughter named Basya or Bisya? Sources please!
The only place in Tanach where the name Basya/Bisya is mentioned is in Divrei Hayamim 1 4:18. Per Koren the pronunciation is Bisya. So how did Bisya become Basya? The Yalkut Shimoni on this Posuk cites the following Medrash. Hashem said to the daughter of Pharoh "Moshe was not your son however you called him your son, therefore though you are not my daughter I will call you my daughter". Thus it follows that Bisya became Basya - the daughter of Hashem.
The confusion seems to be relatively recent. Here is some history from Alexander Beider's A Dictionary of Ashkenazic Given Names (p478-9):
The Ashkenazic female name Basye comes from Batsheva (בת שבע) via hypocorism (and Ashkenazic pronunciation): בת שבע > Basheve > Sheve/Bashe/Basye. Beider points out that the popularity of these variants of Batsheva were "partly due to the existence of a Polish Christian name Basia, a hypocoristic form of Barbara (of Greek origin)." He brings many examples of the name from name corpora.
Regarding the Biblical name, Beider writes that the spelling בתיה "can correspond to two independent names: First, to the biblical בִּתְיָה (1 Chronicles 4:18; a daughter of Pharaoh), pronounced Bisye in Yiddish. Second, it can represent a Hebraized spelling of the Yiddish name Basye, the spelling chosen in order to fit to the above biblical name." Beider brings examples of the Yiddish name Bisye, one of which is spelled ביתייה (which is clearly Bisye, not Basye).
Beider cites a theory that a related name Pese is derived from בַּתְיָה, with the meaning "daughter of God", but rejects it. He notes that the spelling בתיה for Basye is unknown before the 17th century.
Based on this, it seems to me that the pronunciation Basye for the daughter of Pharaoh is a later phenomenon, arising from the spelling בתיה for the name Basye of independent origin (from Batsheva). The pronunciation Bisye for the daughter of Pharaoh has roots both in Tanakh and even in Ashkenazic name lists (where you expect the confusion to be widespread), and should be taken as primary.