Nylon 66 is so named because the

Nylon 66 is made from two different monomers, each containing 6 carbon atoms.
One monomer is adipic acid, which has a chain of 6 carbon atoms.
The other monomer is hexamethylenediamine, which also has a chain of 6 carbon atoms.
The "66" in Nylon 66 indicates that both monomers have 6 carbon atoms.

Let's look at why the other options are incorrect:

(A) Average degree of polymerization of the polymer is 1966: This is incorrect. The degree of polymerization refers to the number of repeating units in the polymer chain, and 1966 is just an arbitrary number.
(C) Number of nitrogen atoms between two carbon atoms are 6: This is incorrect. The number of carbon atoms between the nitrogen atoms is what defines the "66" in Nylon 66.
(D) Polymer was first synthesized in 1966: This is incorrect. Nylon 66 was first synthesized before 1966.