Why is the notation for gain to be \$g_m\$?
Mutual conductance was named by Alan Hazeltine about 1918. Later, sometime in the 1930s, the IRE adopted transconductance as fitting better with other electrical definitions. But mutual conductance continued to be used.

Whereas gain in tube amps related plate/grid voltage gain

\$\mu = dVp / dVg~~\$ and 
\$R_p = dVp / dIp~~~~\$ for apparent plate resistance

thus gain
\$\mu =  g_mR_p\$