Here are three sources, showing both the Lawton name (which then changed to the Layton name) and the fact that this William Lawton/Layton, migrated from Rhode Island to New Jersey:

1.) As I stated above, on 17 Aug 1702, in Middletown, Monmouth, New Jersey, a will was filed in Monmouth County. The will is signed by "William Lawton".

2.) In the book, Salter's History of Monmouth & Ocean Counties, is the following:

Layton, Lawton - William Layton was a "townshipper" among original settlers of Monmouth, 1667. He had town lot number fourteen in Middletown. In 1677, he received a warrant for 183 acres of land from proprietors. Among taxpayers in Middletown township, 1761, was Andre, William, Leonard, Obadiah, Safety, John, and Samuel and son. This family may descend from Thomas Layton who was among original settlers of Portsmouth, Rhode Island, 1638.

3. In the book, This Old Monmouth of Ours, by William S. Horner is the following:

Layton (Lawton) -William Layton, a Baptist, from Rhode Island, was awarded town-lot No. 14. He is supposed to have died in 1708. The name of his wife may have been Violet. His children were William, Thomas, Andrew, John, Samuel and Mary who married a Drummond, probably Gavin.