Of course, since we are talking about approximate solutions, not exact ones, we may want to modify the notion of the condition number a bit. One possible quantity that seems relevant to the "approximate uniqueness" is the following: the least square problem is just about minimization of a quadratic form $Q(x)$ and if $y$ is the solution, then $Q(x)=Q(y)+(A(x-y),(x-y))$. Now, we want to see what is the penalty for going away from the optimal vector. So, both $\frac{\mbox{Tr\,}A}{Q(y)}$ and $\frac{\mu(A)}{Q(y)}$ where $\mu(A)$ is the least eigenvalue of $A$ seem to make sense as measures of such penalty. The higher is this number, the more unique is the approximation. The reason for the denominator is that I wanted to measure the sizes of deviations that change the minimum by certain percentage. You may want to do the absolute error instead, or something else. It may be a good idea to figure out what invariance properties you want from your measure first. For instance, should it be invariant with respect to stretchings or you think that two close points determine a line less precisely than two distant points?