After several years of thinking on this, my favourite hypothesis is a medieval Asian sailor leaving his DNA in a South coast port such as Southampton.
There are two families in southern England that share the same L. During C18 the two surname lines were only 34 miles apart in the area of Berkshire. My surname most likely originated in Sussex, Surrey, Hampshire where it was most common during C16. The south ports along the channel just look likely to be the arrival point. At some point the DNA entered at least two surname lines.
No other L samples in Europe are particularly close to ours. The nearest so far are from Saudi Arabia, Iran, Goa, and Lebanon. Str dating and distribution might suggest it arrived in England sometime between 2000 and 500 years ago.
Of course it could be a Syrian archer in the Roman period, a Byzantine craftsman, an extinct Sephardic line, a Christian or Parsi refugee etc. So many possibilities. But I am always sensitive to the fact that most history only reflects the movements of the rich and powerful. That most individual stories are unrecorded. So I like the idea of a medieval sailor from somewhere in SW Asia moving between ships and ending up leaving a son in an English port.