The iris doesn't finish making its melanin until somewhere between 6 months and 12 months. Some kids are still shifting at 18 months. By age 3, the color is essentially final.
Newborns of European descent are usually born with grey or blue eyes because melanin production hasn't started yet. African and Asian babies are typically born with brown eyes because melanin starts producing earlier in pigmented populations.
What to watch for
If your baby's eyes look hazel or have flecks of brown by 4 months, they're very likely to end up brown. If they're still pure blue at 9 months, they probably stay blue. Green is the rarest — and the hardest to predict.