documentclass[12pt]{article}
usepackage{amsthm}

newtheoremstyle{underlinenonum}% name
{-1.5mm}           % Space above, empty = `usual value'
{}                 % Space below
{}                 % Body font
{parindent}       % Indent amount (empty = no indent, parindent = para indent)
{}                 % Thm head font
{}                 % Punctuation after thm head
{ }                % Space after thm head: newline = linebreak
{noindent{underline{thmnote{#3:}}}}

theoremstyle{underlinenonum}
newtheorem*{subcase}{subcase}
    letmtendsubcaseendsubcase
    renewcommand{endsubcase}{renewcommandqedsymbol{tinyfbox{a}}qedmtendsubcase}

begin{document}
    begin{subcase}[Name of Thing] This is a thing I would like to say.end{subcase}
end{document}