\hyperref with the optional argument can be used to link to labels with arbitray text:
\hyperref[<label>]{<arbitrary text>}
The star form \ref* sets a reference without additional link, since the link is already set by \hyperref.
The following example defines \figref for figure references. The optional argument takes the subfigure number, the mandatory argument the label name without prefix fig:.
Full example:
\documentclass[10pt,a4paper]{article}
\usepackage[T1]{fontenc}
\usepackage{lmodern}
\usepackage{todonotes}
\usepackage{hyperref}
\newcommand*{\figref}[2][]{%
\hyperref[{fig:#2}]{%
Figure~\ref*{fig:#2}%
\ifx\\#1\\%
\else
\,#1%
\fi
}%
}
\begin{document}
\begin{figure}[t]
\missingfigure[figwidth=6cm]{- << Ciel, mon mari >> dit-elle !}
\caption{a: Missing figure, and b: still missing figure.}
\label{fig:missing Figure}
\end{figure}
The figure \figref{missing Figure} contains two subfigures.
Something in \figref[a]{missing Figure} and some other thing in
Figure \figref[b]{missing Figure}.
\end{document}

Version as generic command form via \myref (I do not recommend redefining \ref):
\documentclass[10pt,a4paper]{article}
\usepackage[T1]{fontenc}
\usepackage{lmodern}
\usepackage{todonotes}
\usepackage{hyperref}
\usepackage{twoopt}
\newcommandtwoopt*{\myref}[3][][]{%
\hyperref[{#3}]{%
\ifx\\#1\\%
\else
#1~%
\fi
\ref*{#3}%
\ifx\\#2\\%
\else
\,#2%
\fi
}%
}
\begin{document}
\begin{figure}[t]
\missingfigure[figwidth=6cm]{- << Ciel, mon mari >> dit-elle !}
\caption{a: Missing figure, and b: still missing figure.}
\label{fig:missing Figure}
\end{figure}
The figure \myref[Figure]{fig:missing Figure} contains two subfigures.
Something in \myref[Figure][a]{fig:missing Figure} and some other thing in
Figure \myref[Figure][b]{fig:missing Figure}.
\end{document}
\eqref
A star form \eqref is not provided, but something like (\ref*{...}) can be used instead. Or more precise, \eqref is defined in package amsmath as:
\newcommand{\eqref}[1]{\textup{\tagform@{\ref{#1}}}}
Then \eqrefstar without link can be defined as:
\makeatletter
\newcommand*{\eqrefstar}[1]{\textup{\tagform@{\ref*{#1}}}}
\makeatother
\eqrefstar, a\eqrefwithout link. – Heiko Oberdiek Sep 08 '15 at 11:21