Identify the word or phrase that the underlined appositive phrase renames. astronomy william mitchell maria mitchell a child

Assuming the original sentence is As a child, Maria Mitchell studied astronomy with her father William Mitchell, an amateur astronomer, the correct answer here is William Mitchell. 
An appositive phrase is used to further describe a particular noun next to it, and is usually separated from the rest of the sentence using commas, as is the case here. The appositive phrase an amateur astronomer refers to the father, William Mitchell.