I have found more evidence indicating that his son Thomas Eves (Eves-252) was from Flower/Flore, Northamptonshire. For more details, see that profile ( https://www.wikitree.com/wiki/Eves-252 ):
Land records show his fellow immigrant to Burlington Co., New Jersey (Thomas Olive) was from Wellingbourne, Northamptonshire (and it, like Flower/Flore is near the town of Northampton).
Besides that geographical clue, his children's names also provide a clue. Thomas named three sons John, Daniel, and Samuel, in exactly the same birth order as his brothers (also John, Daniel, and Samuel) born at Flower/Flore, Northamptonshire.
There is no evidence that Thomas Eves came from Evesham, Worcestershire. The family tradition that Evesham Township, New Jersey, was named after that town or after Thomas himself is incorrect. When it was founded in 1688, it was named Eversham (with an "r") Township, and only later was the "r" dropped.