Here are two possible solutions:
pdflatex
You need to load some additional packages:
\usepackage[utf8]{inputenc}
\usepackage[english,greek]{babel}
Then inside the second field of \newacro
, add \selectlanguage{english}
to typeset that word in Latin script.

\documentclass{book}
\usepackage{acronym}
\usepackage[utf8]{inputenc}
\usepackage[english,greek]{babel}
\begin{document}
\chapter*{Ακρωνύμια}
\addcontentsline{toc}{chapter}{Ακρωνύμια}
%place for all the ackronyms
\begin{acronym}[BisGMA]
\setlength{\itemsep}{-\parsep}
\acro{BisGMA}[\selectlanguage{english}\textit{BisGMA}]{Δικλυδικός μεθακρυλικός εστέρας της δισφαινόλης Α}
\end{acronym}
\acl{BisGMA}
\end{document}
xelatex
If you're willing to switch to this engine, you can choose any font you have on your OS that supports both scripts. I chose a random one, but you can change it to a different one.

\documentclass{book}
\usepackage{acronym}
\usepackage{fontspec}
\setmainfont[Script=Greek]{Times New Roman}
\usepackage{polyglossia}
\setdefaultlanguage{greek}
\begin{document}
\chapter*{Ακρωνύμια}
\addcontentsline{toc}{chapter}{Ακρωνύμια}
%place for all the ackronyms
\begin{acronym}[BisGMA]
\setlength{\itemsep}{-\parsep}
\acro{BisGMA}[\textit{BisGMA}]{Δικλυδικός μεθακρυλικός εστέρας της δισφαινόλης Α}
\end{acronym}
\acl{BisGMA}
\end{document}