While this is the sort of question that's probably best brought up in the Random Question Thread, you could use leof (darling), culver (dove), eyas (a young hawk; Romeo uses it towards Juliet in Romeo And Juliet), myn lykyng (the one who gives me pleasure), or my sweeting (my sweet one). If it's directed towards a child, you could also use miting (little mite) or fauntkin (young child, with a diminutive ending).
Out of curiosity, why does it have to be obscure? If it's just there to be "authentic", you're really just better off using a modern version.
Not terribly obscure, but peculiar and old-worldish enough to get the feel across: 'lamb' or 'my lamb' when referring to a child or other 'innocent.'
I've returned from the depths to continue politely irritating the good people of TV Tropes.(◕‿◕✿)Lambkin is the word you're really looking for, there. Still actually used directly translated in Afrikaans, mind: lammetjie. A late lambkin (laatlammetjie) is somebody born well after other siblings. Or, just as a surprise when Mum thought she'd hit the menopause, rather.
I just put that out there, as many phrases forgotten in English turn up in Dutch, Flemish and Frisian. And, visa versa, as well.
From North East England, you get "little sparrow"... or "spuggie". That goes back quite a bit to describe any small child, but girl-children in particular, and spuggie itself used to be far more wide-spread (North of England) for any nondescript, small bird.
I need a medieval term of endearment, something obscure that most people in the present wouldn't have heard of.