Q: what is being "relaxed" in the relaxation method?
A: The relaxation method is an iterative approach to solve the set of linear equations $\sum_{j}M_{ij}v_j-c_i=0$$\sum_{j}A_{ij}u_j-b_i=0$ by relaxing the requirement that the right-hand-side should vanish. Residuals $Z_i$ allow the right-hand-side to be nonzero, $\sum_{j}M_{ij}v_j-c_i=Z_i$$\sum_{j}A_{ij}u_j-b_i=Z_i$, and then the approach consists of iteratively relaxing the $Z_i$'s so that they all settle down on the value 0.
The simplest method successively relaxes to zero the largest residual $Z_{i_0}$ (by adjusting the corresponding $v_{i_0}$$u_{i_0}$).
Concerning the history of the approach, see A historical review of iterative methods. Southwell's early work is described as follows:
Southwell applied relaxation methods to the solution of linear systems arising from the solution of partial differential equations by finite difference methods. Given an initial approximation to the true solution one would record the approximate value $u^{(0)}$ of the solution at each grid point together with the component of the residual vector, $b-Au^{(0)}$ at the point. By inspection of the residuals, one would introduce displacements at one or more grid points and appropriately modify the residuals. The process was continued until all of the residuals became small.