You write: "I have wondered if there is family connection also."
When I posted this query originally, I never doubted that there was a family connection in most instances where "Lyman" was conferred as a baptismal name. What seemed evident (but keep in mind that what seems evident is not always real) was that "Lyman" was conferred disproportionately to most other possible ancestral names. It may be merely that the Lymans had an extremely high birthrate--but having researched other New England families--including the Russells--I doubt that the Lymans had a propriogenitive edge. Certainly "Lyman" has a euphonious quality that "Putnam"--to cite one example--does not possess. There were a number of early Lymans (General Phineas Lyman, for instance) who did loom large in Colonial history. Whatever the answer, it is not a name conferred often today.And until somebody actually does an empirical study of early nineteenth century naming the whole question I posed originally remains speculation--nothing more!