expl3: lowercase a token list

Update 2020-01-14

The LaTeX kernel now makes \\ robust out-of-the-box. When used with the latest expl3 function \text_lowercase:n, this works with no additional adjustments. The comment in the original answer about strings versus text remains valid: you are case-changing text.

Original answer

The function \str_lowercase:n is for making strings lower case, and is for programmatic data not for text. You want \text_lowercase:n. The only issue is that \\ is not engine robust and the implementation of \text_lowercase:n expects 'text' to be either character tokens, things which expand to character tokens or engine-robust commands. We can solve that by locally making \\ engine-robust:

\documentclass{article}

\usepackage{xparse}

\ExplSyntaxOn
\NewDocumentCommand \LowerCase { m }
  {
    \group_begin:
      \cs_set_protected:Npx \\ { \exp_not:o \\ }
      \text_lowercase:n {#1}
    \group_end:
  }
\ExplSyntaxOff


\begin{document}
  \LowerCase{This is my\\Text}
\end{document}

If you make \\ robust globally then everything stays expandable

\documentclass{article}

\usepackage{etoolbox}
\robustify\\
\usepackage{xparse}

\ExplSyntaxOn
\DeclareExpandableDocumentCommand \LowerCase { m }
  { \text_lowercase:n {#1} }
\ExplSyntaxOff

\begin{document}
  \LowerCase{This is my\\Text}
\end{document}

Note that there is no need to store #1 in a token list variable in either case.