How many distinct languages existed in North America in pre-Columbian times?

It is impossible to answer that question since many languages existed and became extinct before there was anyone around to record them - that is the precise meaning of "prehistory" and we can only know about those very ancient times from the archaeological record, which does not preserve language.

It is estimated that in South America there have been many thousands of native languages, many long extinct; there were around 1,500 at first European contact. In Meso-America (Central America) there have been an unknown number of native languages and dialects, certainly several hundreds.

In North America there have been anywhere between 300 and 700 languages and dialects, many of them extinct long before European contact. Many of the archaic languages are completely unknown, such as the language or languages of the Adena, Hopewell and Clovis cultures which may each represent more than one tribe.