I'm also a maternal descendant of Richard Lyman, and you and Todd are correct: No proven royal lineage. In fact, Richard's wife Sarah's surname is unknown. Some secondary sources call her Sarah, daughter of Roger Osborne of Halstead, Kent, but I haven't seen any documentary evidence to support that conclusion.
I'm a paternal descendant of Richard Lyman, discovered only recently while verifying another Revolutionary War Patriot ancestor, John Lyman (1760-1840). I was more than a little surprised to come across the Lyman family history book's claims of Richard's royal roots. Of course, I cast a critical eye and am not surprised that much of it very well may be the result of family legends and wishful thinking. In my genealogical research, I've verified some family "legends," turning them into documented facts, and debunked others, relegating them to the dustbins of myth. Always appreciative of the truth and the honest folks in this thread.