The Swaim family arived in the US from Leerdam, Holland and settled on Staten Island. They were part of the original settlers of Staten Island. Three children came with tme family, Barent, Beleytje and Anthony. Barent and Belytje were children by Thys first wife Peterke jans. Anthony was the son of Sytje. They came to the US aboard the "De St. Jan Baptist" May 9 1661. You'll notice that the surname changed. The story is that the family were protestors. They were working secretly and took the name name Swaim, which stood for "S Working Against I Monarchs." As this was activity was frowned upon by those in power nothing was written. This was is probably just an oral family tradition that may or may not even been true. And of course they were dutch and would not have used English. What were the Dutch words? And would they still have the same first initials? More likely just a coincidence discovered after they arrived.
There is by the way more theories about the name on that web-page http://freepages.genealogy.rootsweb.ancestry.com/~smithhouse/smithg...
You should look at the New Amsterdam series of projects. i hope that your ancestor's profile is already there. Here's a link to the umbrella project: http://www.geni.com/projects/New-Amsterdam. George J. Homs was the mastermind behind the project and it's one of the more successful, collaborative projects on Geni and of interest to hundred of thousands of descendants potentially.
Depends on what you are looking for.
Phone numbers? http://www.detelefoongids.nl/