My surname is Lyon and we like to think we are related to the aristocratic Lyons!
The original royal link is indeed through Robert II. His chamberlain (finance minister in Scotland), Sir John Lyon, married the king's eldest daughter, Jean Stewart, in 1372 as her second husband. They produced one son, another John, from whom the Lyons of Glamis are descended. Sir John Lyon's origins are unclear. He seems to have been the first of the name. However, www.baronage.co.uk
has made the interesting suggestion on the basis of his coat of arms (argent, a lion rampart azure, overall a riband gules) that in an age when surnames were fluid he was a Bruce who changed his name. The arms of the English branch of the Bruces were argent, a lion rampant azure, and, it is argued, no one in those days would adopt a coat of arms to which he was not entitled by blood, particularly when a Bruce occupied the throne (David II 1329-71). A later Lyon duly became Lord Glamis, and in due course his descendants became successively Earls of Kinghorn and then Strathmore.
Where my rather modest Liverpudlian ancestors fit in I don't know!