trema (ddot) does not appear in mathrm mode with newtxmath font

With the default setup you need to use the text accents not math so \textrm{Möbius} but \operatorname gives "log-like" prefix operator spacing, so to get the effect of that, wrap in the primitive \mathop command, so

\mathop{\textrm{Möbius}}

This works:

\documentclass[a4paper,10pt]{article}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{amsmath}
\usepackage{newtxtext,newtxmath}

\DeclareMathOperator{\mobius}{M\mathnormal{\ddot{\mathrm{o}}}bius}

\begin{document}

$\mobius$

\end{document}