The general solution will be of the form $\{x_0+v:v\in V\}$, where $x_0$ is a particular solution and V is an (n-m)-dimensional subspace. One way to minimise the 2-norm of $x_0+v$ is to construct an orthonormal basis $\{e_1,\ldots,e_{n-m}\}$ of V. The closest point to $x_0$ in V is then $v_0 = \sum_{j=1}^{n-m}\langle x_0,e_j\rangle e_j$, and the solution with smallest 2-norm is $x_0-v_0$.