July 10th 1856 at 055 Smiljan, Croatia.

I also found an article which references his birth certificate according to the old calendar as 28. June 1865. On his U.S. citizenship request he put July 15th although it is been speculated that he was hiding from Austrian military draft and was hiding on the mountain for a year. 

It looks like night July 9-10, right after midnight is the time that is most commonly used. 

Tesla's museum page with bio and DOB  http://www.nikolateslamuseum.org/web/page.php?p=11&s=20&l=en

The reason Nicola Tesla was able to create the things he did was that he had extra terrestrial help and used a base 3 or 6 counting system.

That is the most efficient counting system and it gets rid of a lot of troubles. Apparently under a base 6 counting system pie is an integer, and we see it everywhere, 12 star signs, 3 appearing in religion, base 6 way of calculating time etc.

This is what happened at the tower of babel. To stop people becoming too technologically advanced and creating weapons and such to "war against god" the common language, i.e. mathematics, was messed up by giving humans five fingers instead of three or six.

Under a base 3 counting system any technology created is intelligent enough to think for itself. Instead of using 1 and 0 you use 0, 1 and 2!