A function that create function with string manipulation

An approach with stringstrings

\documentclass{article}
\usepackage{stringstrings}
\newcommand\aCustomFunction[2]{%
  \expandafter\def\csname #1\endcsname{#2}%
  \capitalizewords[q]{#1}%
  \expandafter\def\csname\thestring\endcsname{\capitalizewords{#2}}%
}
\begin{document}
\aCustomFunction{dog}{cane}

\dog

\Dog
\end{document}

As it is, the above approach reinvokes \capitalizewords for every instance of \Dog. That can be avoided in the following way, so that \Dog literally contains the string "Cane":

\documentclass{article}
\usepackage{stringstrings}
\newcommand\aCustomFunction[2]{%
  \expandafter\def\csname #1\endcsname{#2}%
  \capitalizewords[q]{#1}%
  \edef\tmp{\thestring}%
  \capitalizewords[q]{#2}%
  \expandafter\edef\csname\tmp\endcsname{\thestring}%
}
\begin{document}
\aCustomFunction{dog}{cane}

\dog

\Dog
\end{document}

Note that either of the approaches will allow indirect syntax such as

\def\blah{this is a test}
\aCustomFunction{cat}{\blah}

\cat

\Cat

However, only the original approach will keep track of things if \blah is redefined mid-stream.

With expl3:

\documentclass{article}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}

\usepackage{xparse}

\ExplSyntaxOn
\NewDocumentCommand{\CustomFunction}{mom}
 {% #1 = command name, #2 = optional initial, #3 = string
  \IfNoValueTF{#2}
   {
    \verdoja_custom_function:ffff
     { \tl_head:n {#1} } { \tl_tail:n {#1} }
     { \tl_head:n {#3} } { \tl_tail:n {#3} }
   }
   {
    \verdoja_custom_function:ffnn 
     { \tl_head:n {#1} } { \tl_tail:n {#1} }
     { #2 } { #3 }
   }
 }
\cs_new_protected:Nn \verdoja_custom_function:nnnn
 {
  \tl_new:c { #1 #2 }
  \tl_set:cn { #1 #2 }{ #3 #4 }
  \tl_new:c { \text_uppercase:n { #1 } #2 }
  \tl_set:cx { \text_uppercase:n { #1 } #2 } { \text_uppercase:n { #3 } #4 }
 }

\cs_generate_variant:Nn \verdoja_custom_function:nnnn { ff , ffff }
\ExplSyntaxOff

\CustomFunction{dog}{cane}
\CustomFunction{elite}[é]{lite}

\begin{document}

\dog

\Dog

\elite

\Elite

\end{document}

If the first character is special, it's necessary to use the optional character (for compatibility with pdflatex, in case XeLaTeX or LuaLaTeX is used the optional argument is not required).

Actually, with the recent additions and refinements to expl3, we can vastly simplify the code.

\documentclass{article}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}

\usepackage{xparse}

\ExplSyntaxOn
\NewDocumentCommand{\CustomFunction}{mm}
 {% #1 = command name, #2 = string
  \verdoja_custom_function:nn { #1 } { #2 }
 }
\cs_new_protected:Nn \verdoja_custom_function:nn
 {
  \tl_const:cn { #1 } { #2 }
  \tl_const:cx { \char_titlecase:N #1 } { \text_titlecase:n { #2 } }
 }

\ExplSyntaxOff

\CustomFunction{dog}{cane}
\CustomFunction{elite}{élite}

\begin{document}

\dog

\Dog

\elite

\Elite

\end{document}

We no longer need the initial to be in an optional argument if “special".