The equation $|\nabla u|=f(u)$ can be recast as $|\nabla v|=1$ in terms of $v=g(u)$, if $f>0$.
So let us suppose $f\equiv1$. To determine $u$, you need a boundary condition $a$ on the boundary $\partial\Omega$ of your domain. Remark that a necessary condition is that $|\nabla_Ta|\le1$, where $\nabla_T$ denotes the tangential gradient.
In general, you don't have a classical solution. Think for instance to the problem in a ball with $a\equiv0$; in which case you have only bad choices, as $u\equiv|x|-1$ or $\equiv1-|x|$. Therefore one must search for Lipschitz solutions that satisfy the equation almost everywhere. To recover uniqueness, one must know where does the boundary value problem come from. In general, the model behind it contains a unilateral condition, often called an viscosity condition. This theory was developed by M. Crandall and P.-L. Lions in the 80's. For the equation above, this yields the viscosity solution
where $\gamma$ runs over the paths from the boundary $\partial\Omega$ to $x$, and $\ell(\gamma)$ is its length. If $\Omega$ is convex, this is just