The French babel module redefines certain bits of the document to conform with French typographical standards and customs. It only does this if it is the document main language (if it is loaded last), but in that case it applies these changes to all parts of the document, even parts written in a different language (this surprised me at first).
Amongst others, the default itemize labels are changed. You can turn this off and return to the standard behaviour with
\frenchsetup{StandardItemLabels=true}
Alternatively, you can modify the labels of only certain levels with
\frenchsetup{ItemLabeli=\textbullet, ItemLabelii={\normalfont\bfseries \textendash}}
Accepted keys are ItemLabeli up to ItemLabeliv.
For even finer control over list environments you can also check out enumitem.
If you had not used French, \renewcommand{\labelitemii}{$\star$} would indeed have worked. But since language redefinitions are loaded close to the end of the preamble this redefinition never survived long enough to take effect. (You can see that \AtBeginDocument{\renewcommand{\labelitemii}{$\star$}} seems to work, but the methods of \frenchsetup should be preferred.)
BTW: My babel tells me to use \usepackage[french]{babel} and not \usepackage[frenchb]{babel}
MWE
\documentclass[a4paper,11pt,oneside]{article}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage[french]{babel}
\frenchsetup{StandardItemLabels=true}
\begin{document}
\begin{itemize}
\item First Level
\begin{itemize}
\item Second Level
\begin{itemize}
\item Third Level
\begin{itemize}
\item Fourth Level
\end{itemize}
\end{itemize}
\end{itemize}
\end{itemize}
\end{document}
