The name in Dutch would have the "van" part of it separated, meaning "from". The Nes or Es is some geographical reference, possibly a river but I am not sure about that.
van Nes and some of the variants van Es or van Nus are fairly common names so you would not be able to get connections unless possibly they were from the same city your birth mother was born in. In the absence of that you would have a better chance of winning the lottery, the Vanness family is simply too big.
If you knew the city you could try just simply grabbing a phonebook and go down the listing and call them as ask if they know of a relative who gave a baby up for adoption in 1960.