I am not sure how specific you want to get.

If you just want a design, the ideal experiment would be:

1. Select healthy subjects (no sleep problems) who are the same in every respect

2. After making sure they are really comparable in the relevant baseline characteristics, randomize them to either receive sleep deprivation or not

3. Induce sleep deprivation in the experimental group, leave the control group alone

4. Test cortisol levels throughout the experiment

5. Measure offspring weight after birth

6. Build a mediation model to see how much of the variance explained by group membership is mediated by cortisol levels.

You might be able to do this in an animal study. DON'T EXECUTE this design with humans though! It would be highly unethical to cause sleep deprivation for pregnant mothers! 

Agree with Dr. Kekecs !

Nevertheless I would add some considerations:

1. to have a human cohort enrolled for the study you maybe could list inclusion criteria for pregnant women already having their own problems with their sleep time (case cohort)

2. besides, you could (easily?) check cortisol in a selected population of so called "normal" pregnant women (control cohort)

3. as much as the fetus regards I would suggest an echocardiographic and Doppler check of the fetal characteristics (would you expect to see differences regarding flows and growth curves between the two populations ?)

4. as much as the newborn regards, my suggestion is to check cortisol on arterial cord blood at birth (maybe you can consider to check corticol in the cord vein as well to complete data)

5. Eventually it remains a huge doubt: how to check neonatal cortisol beyond perinatal adaptation ? Is it that ethic to collect blood for research purposes or would you be able to use leftovers from standard analyses? This question only can have a locally adapted answer.

All the best

Claudio 

P.S.: your idea is definitely amazing