Does $f(x) = |x|$ have a local minimum? I would like to know whether $f(x) = |x|$ has a local minimum at $x = 0$. 
For a local minimum, the first derivative of the function has to be zero. However, in the case of $|x|$, the function is not differentiable at $x = 0$. Will the point (0,0) still be considered as a local minimum?

You have it the wrong way. A function doesn't have to be differentiable (at a point) to attain a minimum or maximum there. Look up the definition of 'minimum' and 'maximum' in your book or course notes; it is (only) about the neighboring function values.
However: if a function is differentiable, then you have a minimum or maximum when the derivative changes sign there (careful: a zero derivative isn't sufficient; e.g. $f(x)=x^3$).
In your example, $|x|$ attains a local (and even a global) minimum at $x=0$, although the derivative doesn't exist there.
