If you are looking for a term specific to seventeenth-century London underworld cant, you might find these two terms from Richard Head, The English Rogue: Described, in the Life of Meriton Latroon (1665) useful:
Lullabie-cheat A Childe
Palliard One whose Father is a Beggar born
According to Francis Grose (writing in 1785) a lullaby-cheat referred more specifically to an infant.
A New Canting Dictionary: Comprehending All the Terms, Ancient and Modern (1725) notes multiple terms for children, many of which undoubtedly date back to the seventeenth century:
BANTLING, a Child
BRAT, a little Child
A CHIP, a Child. A Chip off the old Block; A Son that is his Father's Likeness; more particularly the Son of a Cooper, or one brought up to the same Trade.
CHITTIFACE, a little puny Child.
CLAPPERDOGEON, a Beggar born and bred [presumably normally applied to adults]. ... The Children of these Villains are stil'd PALLIARDS.
FOUNDLING, a Child dropt in the Streets for the Parish to keep.
FUBBS, a loving, fond word used to pretty little Children and Women.
KID, a Child.
KINCHIN, a little Child.
LULLABY-Cheat, a Child.
RUM-Bob, a young Apprentice; also a sharp, sly Trick. Likewise a pretty short Wig.
SOW-CHILD, a Female Child.
SQUEEKER, a Bar-boy; also a Bastard, or any other Child.
TIM, a young Lass.
TIT, a Horse; also a young prim Lass.
Additional Google searches find seventeenth-century instances of bantling (from 1635), brat (from 1642), kinchin (from 1608), and squeeker (from 1692). I did not find an example of kid in the relevant sense from before 1700, but the expression surely existed in the seventeenth century. Interestingly, Nathanial Bailey, Dictionarium Britannicum: Or a More Compleat Universal Etymological English Dictionary (1730) defines (human) kid far more narrowly than the New Canting Dictionary does:
KID a young Goat, Dan[ish] also a young Person trepanned by a Kidnapper.
I'm not sure what to make of the disparity there. But in any case the New Canting Dictionary gives you quite a few options for referring informally to children.