goes
“with her mother”是插入语,可以把这句话看做“Lucy often goes to the zoo with her mother.”所以主语其实是with前面的人,如果前面是两个人以上就用复数形式.但with和and不一样,如果是“Lucy and her mother often ___ to the zoo.”就要用go了.
还有一种either...or\neither...nor什么的就是就近原则,单复数依据临近助词进的主语,比如：Either students or teacher likes the party.Neither Anne nor her parents do the wash.