Create a capitalized macro token using \csname

You have to isolate the first token in #1 from the rest and uppercase it; the fact that \uppercase doesn't expand anything and puts back the token list into the input stream after its operation can be exploited in the following way:

\newcommand{\mycommand}[1]{\mycommandaux#1\relax}
\def\mycommandaux#1#2\relax{%
  \uppercase{\expandafter\gdef\csname #1}#2\endcsname{the macro #1#2 expanded}%
}
\mycommand{abc}
\show\Abc

Of course an empty argument to \mycommand will result in an error.

If you want to provide a definition, say

\mycommand{abc}{Something else}

to be equivalent to \def\Abc{Something else}, just leave out the tokens after \endcsname

\newcommand{\mycommand}[1]{\mycommandaux#1\relax}
\def\mycommandaux#1#2\relax{%
  \uppercase{\expandafter\gdef\csname #1}#2\endcsname
}
\mycommand{abc}{Something else}
\show\Abc

---

The command \uppercase is a bit stranger than other TeX primitives. Indeed the <general text> it requires as argument first does a travel down TeX's stomach to be "regurgitated" after each character token has been transformed using the \uccode vector: if a character has positive \uccode, TeX will transform it into that character (the category code is unchanged). So, for instance, TeX is setup so that \uccode`a=`A and so an a becomes an A after regurgitation. Non character token are left unchanged and no expansion is performed.

Here's the working in slow motion (first version).

1. \mycommand{abc}
   Here #1 is abc

2. \mycommandaux abc\relax
   Here #1 is a and #2 is bc (argument #2 is delimited by \relax, while #1 is undelimited, so the only first token is grabbed)

3. \uppercase{\expandafter\gdef\csname a}bc\endcsname {the macro abc expanded}
   Tex executes the \uppercase and puts back

   \expandafter\gdef\csname A
   

   in the input stream.

4. \expandafter\gdef\csname Abc\endcsname{the macro abc expanded}

5. \gdef\Abc{the macro abc expanded}

Et voilà.

---

A variant with \newcommand instead of \gdef, that makes it easy to define macros with arguments:

\newcommand{\mycommand}[1]{\mycommandaux#1\relax}
\def\mycommandaux#1#2\relax{%
  \uppercase{\expandafter\newcommand\csname #1}#2\endcsname
}
\mycommand{abc}{Something else}
\show\Abc

\mycommand{uvw}[1]{Something else with #1}
\show\Uvw

With \gdef the standard parameter text should be used.

\newcommand{\mycommand}[1]{\mycommandaux#1\relax}
\def\mycommandaux#1#2\relax{%
  \uppercase{\expandafter\gdef\csname #1}#2\endcsname
}
\mycommand{abc}{Something else}
\show\Abc

\mycommand{uvw}#1{Something else with #1}
\show\Uvw

The problem is that \uppercase is not expandable. You can't use it with \expandafter. I would expand the argument first, as good measure, then read the first letter using a second, internal macro, then use \uppercase to change that letter. Finally a second internal macro reads the new uppercase letter and the rest of the name and makes the definition. The original name can also be passed as a third argument if required:

\makeatletter
\newcommand{\mycommand}[1]{% 
    \begingroup
    \edef\@tempa{#1}%
    \expandafter\endgroup\expandafter\@mycommand\@tempa\@nnil{#1}%
}
\def\@mycommand#1{%
    \uppercase{\@@mycommand{#1}}%
}
\def\@@mycommand#1#2\@nnil#3{%
    \global\@namedef{#1#2}{the macro #3 expanded}%%
}
\makeatother
\mycommand{abc}
\show\Abc

This is a good example of why LuaTeX is useful.

\def\defineuppercase#1%
    {\ctxcommand{defineuppercase("#1")}}

\startluacode
  function commands.defineuppercase(s)
      local upper = string.gsub(s, "^%l", string.upper)
      context.setgvalue(upper)
  end
\stopluacode

\defineuppercase{one}{Something}

\show\One
\end

I am using ConTeXt MkIV, but the code can be easily translated to LaTeX as well.