Collaborations and out of character discussions welcome here!
A word on the setting and how the adventure began which might give you somewhere to start on backstories (though you should feel free to be from anywhere on Golarion).
Dr. M's private tropical island is located in Mediogalti (see map in the Pirates Inner Sea Guide) near the perpetual hurricane Eye of Abendego. You will be able to deduce this from the circumstances surrounding your arrival...
Before awakening in your predicament, your party was on the ship Fairwinds bound for the lawful capital city of Ilizmagorti with a client. You met this client, (British-like) aristocrat Count Owain Mairchild in his native land of Andoran. (Perhaps in the Andoran capital your adventuring party has rented an office space?)
Shipping tycoon Count Mairchild came to you asking for help finding his lost fifteen-year-old son, Davey. (Davey had stolen a magic item of considerable power and stowed away on a ship belonging to the Patriotic Pirate-Hunters of Andoran. His ambition was to prove he was ready to join their ranks. He'd left a note to his father explaining this.) But the Pirate-Hunters' ship returned without him.
The Count handled all your travel arrangements and accompanied you on his own ship. Your host's manner has been jovial, airily twirling his handlebar moustache and humming sea chanties on the voyage. Given the chance he boasts non-stop about his only son. He seems completely confident that Davey will be recovered, and tells you your role will be that of investigator and bodyguard. As a reward he has already given you a Type I bag of holding to share (Core Book p. 500).