What is the “eighth day” in Scripture?

Answer:

It is just a way to say, "At the start of the following week."

"For seven days you shall present an offering by fire to the LORD. On the eighth day you shall have a holy convocation and present an offering by fire to the LORD; it is an assembly. You shall do no laborious work" (Leviticus 23:36).

"He read from the book of the law of God daily, from the first day to the last day. And they celebrated the feast seven days, and on the eighth day there was a solemn assembly according to the ordinance" (Nehemiah 8:18).

The start of the count does not necessarily have to begin with the first day of the week. For example, circumcision was done on the eighth day after birth (Luke 1:59).