what is the problem? why not can a frame with round corner?

documentclass{article}

    usepackage[framemethod=TikZ]{mdframed}

    usepackage{lipsum}

    usepackage{float}

    usepackage{copyrightbox}

    makeatletter

    renewcommand{CRB@setcopyrightfont}{%

        footnotesize     %footnotesize  normalsize

        color{black}   % <--- you can select black because

                    }

    makeatother

    begin{document}



      begin{figure}[H]

                centering

                caption{Process}

                label{copia} 

                copyrightbox[b]{

      begin{mdframed}[roundcorner=10pt]

      includegraphics[scale=0.8]{example-image}

      end{mdframed}

      {source: asd }}

    end{figure}



    end{document}

There's a couple of things going on.

Firstly, you have your braces in the wrong place for the copyrightbox macro.

You have: copyrightbox[b]{framesource: asd}}, but you should have copyrightbox[b]{frame}{source: asd}.

Secondly, copyrightbox seems not to like the frame as its image argument. You can work around this by putting it a parbox.

MWE

documentclass{article}

usepackage[framemethod=TikZ]{mdframed}

usepackage{float}

usepackage{copyrightbox}

makeatletter

renewcommand{CRB@setcopyrightfont}{%

  footnotesize

  color{black}}

makeatother

begin{document}

begin{figure}[H]

  centering

  caption{Process}

  label{copia} 

  copyrightbox[b]{%

    parbox{linewidth}{%

      begin{mdframed}[roundcorner=10pt]

        centering

        includegraphics[scale=0.8]{example-image}%

      end{mdframed}%

    }%

  }{source: asd}

end{figure}

end{document}

If you are okay with not using mdframed, I have a solution with tikz.

documentclass{article}

   usepackage{tikz}

   usepackage{float}

   usepackage{copyrightbox}

    begin{document}

      begin{figure}[H]

                centering

                caption{Process}

                label{copia} 

                copyrightbox[b]{tikznode[draw,inner sep=5mm, rounded corners=10pt]{includegraphics[scale=0.8]{example-image}};}

            {source: asd }

    end{figure}

    end{document}