How to print months in French with BibTeX?

You only have to add one of the following sections to your .bib file:

English

@STRING{ jan = "January" }
@STRING{ feb = "February" }
@STRING{ mar = "March" }
@STRING{ apr = "April" }
@STRING{ may = "May" }
@STRING{ jun = "June" }
@STRING{ jul = "July" }
@STRING{ aug = "August" }
@STRING{ sep = "September" }
@STRING{ oct = "October" }
@STRING{ nov = "November" }
@STRING{ dec = "December" }

French

@STRING{ jan = "Janvier" }
@STRING{ feb = "F{\'e}vrier" }
@STRING{ mar = "Mars" }
@STRING{ apr = "Avril" }
@STRING{ may = "Mai" }
@STRING{ jun = "Juin" }
@STRING{ jul = "Juilliet" }
@STRING{ aug = "Ao{\^u}t" }
@STRING{ sep = "Septembre" }
@STRING{ oct = "Octobre" }
@STRING{ nov = "Novembre" }
@STRING{ dec = "D{\'e}cembre" }

German

@STRING{ jan = "Januar" }
@STRING{ feb = "Februar" }
@STRING{ mar = "M{\"a}rz" }
@STRING{ apr = "April" }
@STRING{ may = "Mai" }
@STRING{ jun = "Juni" }
@STRING{ jul = "Juli" }
@STRING{ aug = "August" }
@STRING{ sep = "September" }
@STRING{ oct = "Oktober" }
@STRING{ nov = "November" }
@STRING{ dec = "Dezember" }

You don't indicate which bibliography style file you use, so I'll assume that it's the "default" file, viz., plain.bst. If so, the problem you're facing is that this file contains the instruction

MACRO {jun} {"June"}

Thus, the string "jun" in your bib file will be typeset as "June" unless there's an explicit override. You can provide this override by inserting the instruction

@string{ jun = "Juin" }

at the top of your bib file.

Separately, I'd also recommend that you change the line

howpublished = {URL: \texttt{http://www.awebsite.com/}}

to

howpublished = {URL: \url{http://www.awebsite.com/}}

and load the url package (which provides the command \url). The advantage of taking this approach is that the url package can (almost) always choose a suitable line break for long and complicated url strings, letting you avoid massively underfull or overfull lines.

Taking these observations together, your .bib file (called, say, urlcite.bib) should contain:

@string{ jun = "Juin" }
@Misc{url-site,
  key          = {site},
  title        = {\textit{site}},
  month        = jun,
  year         = 2012,
  howpublished = {URL: \url{http://www.awebsite.com/}}
}

A full MWE might then look like this:

\documentclass{article}
\usepackage{url}          % for the \url command
\bibliographystyle{plain} % or whatever style file you prefer
\usepackage[french]{babel}
\begin{document} 
\cite{url-site} 
\bibliography{urlcite}    % if bib entries are in the file "urlcite.bib"
\end{document}