One may prefer to write your equation in the form
$$
v_t = \sigma(x) |v_x|
$$
where $\sigma(x) \ge 0$ for $x \in R$. You need to define an initial condition e.g.
$$
v(x,0) = v^0(x) .
$$
Let me skip a treatment of boundary conditions (you hardly can solve the equation numerically on an unbounded domain). Note that the exact solution $v(x,t)$ for a fixed $x$ is non-decreasing function in time.
The simplest (first order accurate) numerical scheme is based on Rouy-Tourin method. You aim to find the values $v_i^n$ that approximate the unknown exact values $v(x_i,t^n)$ for discrete points $x_i \in R$ and $t^n > 0$. Let me suppose uniform steps $h=x_{i+1}-x_i$ and $\tau=t^{n+1}-t^n$.
Firstly, $v_i^0 = v^0(x_i)$. Furthermore, the scheme can be formally written as follows
$$
v_i^{n+1} = v_i^n + \tau \sigma(x_i) |\partial_x v_i^n|
$$
where a proper finite difference $\partial_x v_i^n$ (backward or forward one sided) shall be used to approximate $\partial_x v(x_i,t^n)$.
To make the story short such appropriate approximation is based on the fact that the solution $v$ for a fixed $x$ is non-decreasing, therefore one has to choose one sided finite difference depending which of three values, $v_{i-1}^n$, $v_i^n$, $v_{i+1}^n$ is maximal. Namely
$$
\partial_x v_i^n = (v_{i+1}^n - v_i^n)/h \,\,\hbox{ if } \,\, v_{i+1} \ge \max\{v_i^n,v_{i-1}^n \}
$$
or
$$
\partial_x v_i^n = (v_{i}^n - v_{i-1}^n)/h \,\,\hbox{ if } \,\, v_{i-1} \ge \max\{v_i^n,v_{i+1}^n \}
$$
or
$$
\partial_x v_i^n = 0 \,\,\hbox{ if } \,\, v_{i} \ge \max\{v_{i-1}^n,v_{i+1}^n \}
$$
There are other first order accurate methods, e.g. Osher-Sethian scheme, but this one is considered ``less diffusive''.
The stability of scheme can be proved if all local Courant numbers
$$
C_i = \tau \sigma(x_i) / h
$$
are less or equal to one. In fact in such case the discrete minimum and maximum principle is fulfilled for numerical solution (e.g. no unphysical oscillations).
As a reference I would recommend the book of Sethian or/and the book of Osher-Fedkiw, it is easy to find the exact references (P.S. see level set methods).
P.S. I was asked about stability in comments. In fact, as I mentioned above, one can prove even more. The scheme can be written equivalently as
$$
v_i^{n+1} = (1 - C_i) v_i^n + C_i \max \{ v_{i-1}^n, v_i^n, v_{i+1}^n\}
$$
so for $C_i \le 1$ the value $v_i^{n+1}$ is the convex combination of two values, consequently
$$
v_i^{n+1} \le \max \{v_{i-1}^n,v_i^n, v_{i+1}^n\} \,.
$$
