What is the meaning of the word root "aceto"?
The root "acet-", according to wiki, is derived from latin acētum, meaning vinegar, i.e. diluted acetic acid. Consequently, direct derivatives of acetic acid, like acetamide and acetonitrile are called so. Also, acetone called so because it can be produced from acetic acid (directly or by dry distillation of calcium acetate)

What isn't obvious is why acetonitrile  is considered a derivative of acetic acid and why nitriles in general are considered derivatives of carboxilic acids. Partially it's tradition and partially it's because compounds with carbon skeleton and oxidation state of carbon atoms unchanged are often rather easy to convert into each other.