What you're running into here is a scoping issue. In (La)TeX, you can introduce scopes with { ... }, \bgroup ... \egroup, or \begingroup ... \endgroup. The former two are roughly the same, as \bgroup and \egroup are defined by \let\bgroup{ and \let\egroup}; the last one is slightly different. But the scoping property is the same: any commands, boxen, etc., created or modified within those scopes are not visible outside. And in LaTeX, all environments \begin{env} ... \end{env} implicitly wrap their contents in \begingroup ... \endgroup. This means that your\sbox{\somebox}{Some text} modification is only visible until the \end{itemize}; after that, the modification is undone. To get around this, prepend any command like \newcommand, \def, \newsavebox, \sbox, etc., with \global, which forces the definition to take place at the global scope and be visible everywhere.
Also, to use quotes in (La)TeX, write ``double quoted'', ``double quoted", or `single quoted'; the " character is only for closing quotes, not opening quotes. Putting this all together gives you the revised snippet
\documentclass{article}
\begin{document}
\newsavebox{\somebox}
\begin{itemize}
\item hello1
\item hello1 \global\sbox{\somebox}{Some text}
\end{itemize}
This should show something, and in fact does: ``\usebox{\somebox}''
\end{document}