The final -s in the first person appeared around the XIV century. Before this, in old French, you would say “je/jou/jo sui”.

I’d say that the final -s came for euphony (phonetic link with vowels) and appearance reasons by analogy to the Imparfait final -s, that was popularized by Ronsard after the huges changes that it as undergone.

I would add that you could find potential avenues of investigations in old French books as Perceval ou le Roman du Graal (around XII century) in which you would not see any final -s : For example, “Chevaliers sui.” (1st section) which mean “I am a knight” and would be nowadays translated as “Je suis un chevalier”. If you read a translated version of this book (old french => “new” french) you will probably encounter something like “Un chevalier” as “Chevalier sui” answer a question and “Un chevalier” can be used to translate this answer to new french while staying as close as possible from the text. If you read more recent books you will see “suis” with the final -s (by more recent I mean XVI century or even XV).