For you
Photo ID, insurance card, birth preferences (3 copies).
Lip balm, hair ties, comfy robe, slippers with grip, two going-home outfits one size up.
Phone charger with extra-long cable, snacks, electrolyte drinks.
Adult diapers, peri-bottle (the hospital's is fine too), nipple balm.
For baby
Going-home outfit (newborn AND 0–3 mo, in case of size).
Installed, inspected car seat.
Swaddle, mittens, a soft hat.
For partner
Change of clothes, toiletries, snacks, phone charger, blanket and pillow.