jmc

Since the GCD of $m$ and $n$ is 6, $m=6x$ and $n=6y$ for some integers $x$ and $y$. Note that minimizing $m+n=6x+6y=6(x+y)$ is equivalent to minimizing $x+y$.

The LCM of $m$ and $n$ is $126=2\cdot 3^2\cdot 7=6\cdot 3\cdot 7$, so one of $x$ and $y$ is divisible by 3 and one is divisible by 7. Then we can minimize $x+y$ by setting $x$ and $y$ to be 3 and 7 in some order. Therefore, the least possible value of $m+n$ is $6(3+7)=\boxed{60}$.