I've done some research and apparently it comes from the name "Eneko", which was a popular name in the 17th century and still is nowadays. As explained in the previous reply, "-koetxea" or "-etxea" means "the house of", so (e)Nekoetxea would mean "the house of Eneko". This is the only explanation I've found.
However, I have another hypothesis. "Neko" could mean "nireko", "niko" (the same way "nere" is "nire"). It could be translated as "my", and since "etxea" means "house" or "home", I think Nekoetxea could be something like "the house of me" or "my house". It doesn't really make sense and I have no reliable proof that could corroborate or demonstrate it but I just wanted to share my thoughts.
What do you think?