What is the common mode in common mode input voltage range?
Common mode voltage is distinguised from differential mode voltage. 

The differential mode voltage for two inputs is how different they are. That is \$V_1-V_2\$.

The common mode voltage is the part of the voltage that is the same for both, that is, the part that they have in common. As you say, the formula is \$\dfrac{V_1+V_2}{2}\$.

We can make this more mathematical by noticing that with these definitions

\$V_1 = V_c + V_d / 2\$

and 

\$V_2 = V_c - V_d / 2\$.

You can see that the common-mode term is common between these two equations.