How to define a macro to create a new macro with a name passed as its argument?

Use \csname #1\endcsname which must be expanded before \newcommand using \expandafter:

\newcommand\create[1]{%
\expandafter\newcommand\csname #1\endcsname{My name is #1}}
% usage: \create{foobar}

If you want to pass the macro as control sequence instead, i.e. \foobar instead of foobar then you need to turn it into a string and remove the backslash:

\makeatletter
\newcommand\create[1]{%
\expandafter\newcommand\csname\expandafter\@gobble\string#1\endcsname{My name is #1}}
\makeatother
% usage: \create{\foobar}

There is also the \@namedef macro which is defined as \expandafter\def\csname #1\endcsname, so you can use it as:

\newcommand\create[1]{\@namedef{My name is #1}}

The etoolbox package also provides a basically identical, but robust macro called \csdef. For both you can provide a parameter text, e.g. for arguments direct after the name argument: \csdef{name}#1#2{some code with two arguments #1 and #2} (the # have to be doubled inside another macro).

\documentclass{article}

\newcommand\create[1]{%                                                         
\expandafter\def\csname #1\endcsname{My name is #1}}

\begin{document}
\create{test}
\test
\end{document}

If you're going to be doing lots of this sort of thing, you can create a bunch of macros in one go from a comma separated list with some etoolbox magic:

\usepackage{etoolbox}
\newcommand\create[1]{%
\expandafter\newcommand\csname #1\endcsname{My name is #1}}
\newcommand\createlist{%
\let\do\create
\docsvlist
}

The argument of \createlist should be a list like: \createlist{foo,Foo,magic,jabberwocky} and it will create \foo,\Foo,\magic,\jabberwocky