Almost all roads containing lists lead to enumitem
. You can use enumitem
to tweak the list as you want.
\documentclass{amsart}
\usepackage{enumitem}
\setlist[enumerate,1]{label=\arabic*$^\circ$,ref=\arabic*$^\circ$}
\begin{document}
\begin{enumerate}
\item one
\item two \label{item:some}
\item three
\end{enumerate}
\ref{item:some}
\end{document}

If you insist on using your method, you have to renew labelenumi
which is responsible for printing the item numbers.
\documentclass{amsart}
\renewcommand{\theenumi}{\arabic{enumi}$^\circ$}
\renewcommand{\labelenumi}{\theenumi} %% this needed
\begin{document}
\begin{enumerate}
\item one
\item two \label{item:some}
\item three
\end{enumerate}
\ref{item:some}
\end{document}