There are two issues you need to consider here I suspect: the font shape itself, serif vs sans serif; and the font apparent size (MS font in particular tend to look much bigger than other fonts.
For the first issue, Aerlinthe is indeed correct, the SIL fonts seem to be a good bet, especially Gentium which looks very similar to Minion Pro.
For the second problem, I suggest using the fontspec option Scale=MatchLowercase. this will make sure that all font used will match in size.
A demonstration of this is
\documentclass[a4paper,12pt]{article}
\usepackage{fontspec}
\setmainfont{Minion Pro}
\begin{document}
% Define a font capable of rendering IPA (phonetic) symbols
\newcommand{\phoneticFont}{Arial Unicode MS}
\newcommand{\phonetic}[1]{ {\fontspec{\phoneticFont} #1} }
\newcommand{\phoneticScaled}[1]{ {\fontspec[Scale=MatchLowercase]{\phoneticFont} #1} }
was formerly known under various European names, such as Budschaja in German,
Bugia or Bougia in Italian, Bougie [\phonetic{bu'ʒi}] in French or Bidjaya.
was formerly known under various European names, such as Budschaja in German,
Bugia or Bougia in Italian, Bougie [\phoneticScaled{bu'ʒi}] in French or Bidjaya.
\renewcommand{\phoneticFont}{Gentium}
was formerly known under various European names, such as Budschaja in German,
Bugia or Bougia in Italian, Bougie [\phonetic{bu'ʒi}] in French or Bidjaya.
was formerly known under various European names, such as Budschaja in German,
Bugia or Bougia in Italian, Bougie [\phoneticScaled{bu'ʒi}] in French or Bidjaya.
\end{document}
which produce the following result

As you can see, irrespective of the font used the scaled version fits much more nicely with the text.
So for your actual document the file should look something like
\documentclass[a4paper,12pt]{article}
\usepackage{fontspec}
\setmainfont{Minion Pro}
% Define a font capable of rendering IPA (phonetic) symbols
\newfontfamily{\phoneticfamily}[Scale=MatchLowercase]{Gentium} %replace "Gentium" with the font of your choice
\DeclareTextFontCommand{\phonetic}{\phoneticfamily}
\begin{document}
was formerly known under various European names, such as Budschaja in German,
Bugia or Bougia in Italian, Bougie [\phonetic{bu'ʒi}] in French or Bidjaya.
\end{document}
tipapackage. – You Mar 07 '12 at 01:34