Nearly a year since i started this discussion, so long in fact that i had forgotten it's existence and was about to start a new discussion on the same baptism!
The amount of recent activity on the project had drawn my attention back to it and this same entry (2nd on the list) again drew my attention.
I still think my original feeling was correct, that the witnesses are the key, Cornelis Lamberts Cool is the maternal grandfather. Which makes the unnamed mother one of his daughters. On Geni he has two daughters Pietertje and Aeltje but i have found numerous online references to a daughter named Aechtje some of which conflate her with Pietertje. Obviously online trees are not reliable sources but they are good sign posts.
Interestingly i have also found a well written paper that explains why Pietertje was not married to Claes Janszen van Purmesendt and this matches my theory from January! I love being right :)
So far i have found the following "rumours":
Claes Jansen Rust Van Emden who married Aechtje Cornelissen Cool. Their children were Maryken, Arent b. ca 1642, and Lucas b. ca 1645.
and
Aechtje(Pieterje?) Cornelis, born in Holland, m Claes Jansen (Backer) van Emden
The post on Rootsweb which set me on this trail (this time around) was
"Cool Descendants *think* that the first wife of Cornelis was named Marritje, Maria or Maryken. This is based on the fact that all three of her children; Aeltje, Aechtje and Lambert each named a child with a variance of this name:
* Lambrt's dau:1642 Oct 05; Lambert Lambertszen; Maryken; Jan Pieterszen Roos, Marie Thomas, Tryntie Claes.
* Aeltje's dau: 10 Apr 1644; Gerrit Wolfertszen; Marritje; Wolferts
Gerritszen, Pieter Wolfertszen; Tryntje Huygens, Marritje Philips.
* Aechtje's dau: Claes Janszen, Maryken; sponsers: Corn. Lambertszen Cool, Pieter Wolphertszen, Burger Joriszen Smit, Neeltje Wolperts and Aaltje Cornelis. "
I will see what evidence i can dig up, this post is purely to focus my thoughts.
PS: it is astounding the number of MyHeritage users that list Cornelis Lamberts Cool's birth place as 1588 - Brooklyn NY USA!!!