wait.

why is "минуточку" automatically in the accusative when telling someone to wait?

Also "секундочку" etc...

It’s an ellipsis from «Подожди(те) минуточку» ‘wait a minute’.

Минута прошла. (nominative -- subject of intransitive verb пройти)
A minute passed.

Я провел минуту слушая грызение своей крысы, Рапунцели. (accusative -- direct object of transitive verb провести)
I spent a minute listening to the gnawing of my rat, Rapunzel.

(I only used this example because I think the phonetic correspondence between the noun крыса ("a rat") and the verb грызть ("to gnaw") is a delightful example of two words coming from the same onomatopoeic origin.)