Most biblatex styles use the name format sortname to print the author names in the bibliography, so we can redefine that format.
In particular sortname is the name format used for the authoryear- and authortitle-based bibliography styles (including the verbose family). If you want a similar effect for numeric and alphabetic styles you should redefine the name formats author, editor and translator directly.
First we define a new format: first-last-bold, it is based on first-last, but with an added \mkbibbold{
\DeclareNameFormat{given-family-bold}{\mkbibbold{%
\ifgiveninits
{\usebibmacro{name:given-family}
{\namepartfamily}
{\namepartgiveni}
{\namepartprefix}
{\namepartsuffix}}
{\usebibmacro{name:given-family}
{\namepartfamily}
{\namepartgiven}
{\namepartprefix}
{\namepartsuffix}}%
\usebibmacro{name:andothers}}}
similarly one could have
\DeclareNameFormat{family-given-bold}{\mkbibbold{%
\ifgiveninits
{\usebibmacro{name:family-given}
{\namepartfamily}
{\namepartgiveni}
{\namepartprefix}
{\namepartsuffix}}
{\usebibmacro{name:family-given}
{\namepartfamily}
{\namepartgiven}
{\namepartprefix}
{\namepartsuffix}}%
\usebibmacro{name:andothers}}}
or
\DeclareNameFormat{family-given/given-family-bold}{\mkbibbold{%
\ifnumequal{\value{listcount}}{1}
{\ifgiveninits
{\usebibmacro{name:family-given}
{\namepartfamily}
{\namepartgiveni}
{\namepartprefix}
{\namepartsuffix}}
{\usebibmacro{name:family-given}
{\namepartfamily}
{\namepartgiven}
{\namepartprefix}
{\namepartsuffix}}%
\ifboolexpe{%
test {\ifdefvoid\namepartgiven}
and
test {\ifdefvoid\namepartprefix}}
{}
{\usebibmacro{name:revsdelim}}}
{\ifgiveninits
{\usebibmacro{name:given-family}
{\namepartfamily}
{\namepartgiveni}
{\namepartprefix}
{\namepartsuffix}}
{\usebibmacro{name:given-family}
{\namepartfamily}
{\namepartgiven}
{\namepartprefix}
{\namepartsuffix}}}%
\usebibmacro{name:andothers}}}
then we tell biblatex to use the new format
\DeclareNameAlias{sortname}{given-family-bold}
This last step might have to be
\DeclareNameAlias{author}{given-family-bold}
\DeclareNameAlias{editor}{given-family-bold}
\DeclareNameAlias{translator}{given-family-bold}
with some styles.
MWE
\documentclass{article}
\usepackage[style=verbose-ibid,backend=biber]{biblatex}
\addbibresource{biblatex-examples.bib}
\DeclareNameFormat{given-family-bold}{\mkbibbold{%
\ifgiveninits
{\usebibmacro{name:given-family}
{\namepartfamily}
{\namepartgiveni}
{\namepartprefix}
{\namepartsuffix}}
{\usebibmacro{name:given-family}
{\namepartfamily}
{\namepartgiven}
{\namepartprefix}
{\namepartsuffix}}%
\usebibmacro{name:andothers}}}
\DeclareNameAlias{sortname}{given-family-bold}
\begin{document}
\cite{cicero,wilde,baez/article,knuth:ct,aristotle:physics}
\printbibliography
\end{document}

The answer has been updated to conform with the new name format in version 3.3, see Biblatex 3.3 name formatting.
\mkbibboldinstead of\tetxbf. – moewe May 18 '14 at 12:34beamer, I had to squeeze this inside amakeatletterandmakeatotherjob. TheAtBeginBibliographywasn't working. Of all the answers, this is the only one that worked for me! – PatrickT Jul 03 '18 at 13:59\mkbibnamelasthas become\mkbibnamefamily, apparently. – PatrickT Jul 05 '18 at 20:18\mkbibnamefirsthas become\mkbibnamegiven;\mkbibnameaffixis\mkbibnamesuffixnow. Note that this solution bolds all names (for example also editors of@incollections) which may or may not be desired. – moewe Jul 05 '18 at 20:27author, for some theeditor). It should work for most standard styles, but if you use APA style that probably won't work. In that case I suggest you ask a new question. – moewe Jul 06 '18 at 07:54biblatex-apaif I have to follow the APA style - or a derived style with tiny modifications. Usually the standard styles are much easier to customise and modify becausebiblatex-apahas to work hard to implement the APA guidelines (the same goes for a specialist styles,biblatex-chicago,biblatex-mla, ...).style=authoryearor its derived styles might be a good start. – moewe Jul 06 '18 at 08:31