This is very easy: just build up a list of strings and access them with \coordindex. To this end we define a comma-separated list of strings,
\edef\mylst{"An arbitrary string","String","Custom label","Not this data"}
where the first entry (which has internally index 0) will be used for the first node, the second entry for the second node, and so on. Make sure that the list as at least as many entries as nodes that exist.
\documentclass{article}
\usepackage{pgfplots}
\pgfplotsset{compat=1.16}
\begin{document}
\begin{figure}
\begin{tikzpicture}
\edef\mylst{"An arbitrary string","String","Custom label","Not this data"}
\begin{axis}[width=12cm,
ybar,
enlargelimits=0.15,
legend style={at={(0.5,-0.15)},
anchor=north,legend columns=-1},
ylabel={Speedup},
xlabel={\# of Model Elements (millions)},
symbolic x coords={1m,1.5m,2m,4m},
xtick=data,
nodes near coords=\pgfmathsetmacro{\mystring}{{\mylst}[\coordindex]}\mystring,
nodes near coords align={vertical},
]
\addplot coordinates {(1m,92.021) (1.5m,235.809) (2m,276.824) (4m,340.847)};
\end{axis}
\end{tikzpicture}
\caption{Results}
\label{fig:mycaption}
\end{figure}
\end{document}
The strings are a bit too long. Are you OK with using multiple lines for them?
\documentclass{article}
\usepackage{pgfplots}
\pgfplotsset{compat=1.16}
\begin{document}
\begin{figure}[htb]
\centering
\begin{tikzpicture}
\edef\mylst{"An arbitrary string","String","Custom label","Not this data"}
\begin{axis}[ymax=370,
ybar,
enlargelimits=0.15,
legend style={at={(0.5,-0.15)},
anchor=north,legend columns=-1},
ylabel={Speedup},
xlabel={\# of Model Elements (millions)},
symbolic x coords={1m,1.5m,2m,4m},
xtick=data,
nodes near coords style={font=\sffamily,align=center,text width=4em},
nodes near coords=\pgfmathsetmacro{\mystring}{{\mylst}[\coordindex]}\mystring,
nodes near coords align={vertical},
]
\addplot coordinates {(1m,92.021) (1.5m,235.809) (2m,276.824) (4m,340.847)};
\end{axis}
\end{tikzpicture}
\caption{Results}
\label{fig:mycaption}
\end{figure}
\end{document}

\mystringdoes? – Sina Madani Feb 11 '20 at 17:25babelpackage. In that case you need to load\usetikzlibrary{babel}. If this does not work, add a full compilable minimal working example to your question, my answer works. – Feb 11 '20 at 17:39