We've seen in 'Voyage To The Bottom Of The Sea' that sea-faring men were already in the family tree of Admiral Harriman Nelson. One of his ancestors was Captain Seamus O'Hara Nelson (also played by Richard Basehart in the episode "The Haunted Submarine").
And here at "Inner Toob", I've posited the theory that Aron Sligo, another former sea captain (seen in his eponymous episode of 'Gunsmoke'), was another of his forebears.
I'm now making the claim there is yet another sailor in the Nelson family tree, who served in His Majesty's fleet back in the 1720s.
Stationed in the Bahamas, Lt. Edward Beamish wasn't exactly in the same mold as Admiral Nelson or Captains Sligo and O'Hara Nelson. Without much experience, Beamish was put in charge of the garrison at New Providence after Governor Woodes Rogers was summoned back to England. Headstrong, impetuous, but woefully outclassed by such foes as Blackbeard, Beamish struck up an alliance with Captain Dan Tempest who helmed the Sultana and who was a reformed pirate pardoned by Rogers.
Lt. Beamish was played by Peter Hammond, not Richard Basehart, but I think there is still a strong case to be made that the powerful genetics of the Nelson lineage can be found in him.
But when it comes to his soul, he's more of a Major Frank Burns from 'M*A*S*H' or a Brother Jerome of 'Cadfael'. So it's not likely he could have reincarnated to be Harriman Nelson, not without a lot of other incarnations along the way to improve his karma.
I'll just stick with the claim that Edward Beamish was a member of the Nelson family tree, thanks.