Great find Nick, and a good connection upon reading the old document.
If correct your killing oral and traditional history naming conventions !!
While the English just took control of Niew Amsterdam /New York at the time of this document, the whole area was still Dutch, language and customs. A dutch translation of “greate Gunn Hill” as written in the document would be ‘grote kanonheuvel’. So it looks like a direct translation from the Dutch to English.
As to why the name Gun Hill” ? if I had to guess It’s due to the rough geographical shape of the hill . I can imagine back then the view of the hill from a side could look like a cannon. It’s not common for hills and mountains to be named by what they appear to look like. There are many mountains named ‘sugar loaf’. In our area up the Hudson opposite Bear ountain is Anthony’s nose. Traditioun has it named for Anthony DeHooges who along with Adrian Vander Donck were officers of the Dutch patoonship Rensselaerwyck of Kiliaen Van Rensselaer. Looking at the map you can see the long ridge in the form of a cannon. Again good research in making the connection for a new meaning of a familiar local name.