I made a lot of baby-stuff lately.
I used synthetic fibers, cotton (just a few things), and special fibers like bamboo yarn (very soft!).
In my opinion you should not use the "staight up acrylic" stuff. When you knit that up it makes "screetch sounds" and does not feel plesant.
some wools will scratch and are no go for baby skin.
Any yarn that makes a pleasantly feeling fabric is fine for baby stuff.
Then comes the next question: Things need to be washed. And not everybody knows how to treat wool, so unless you know that it will get propper care, don't use sensitive stuff.
And what kind of piece do you make? a summer hat will be different from a winter hat. A summer dress in angora is hard to imagine, a winter coat in cotton as well. You know what I mean?
Synthetic and wool both keep warm. Good or bad, whatever the piece will be. Then they just feel different, and a densly knit acrylic might be sweaty.
cotton is breathable but can get very stiff. Thin cotton and larger needles are good, though.
My best bet is to go to the shop and feel the yarn. Then I know.
One more thing, whatever you use:
I always give the yarn label away with the piece and tie a bit of yarn to the lable as well as writing the piece on it. That helps the mother of the baby to know the treatment and material.