Parents often find that their babies enjoy sucking on their fingers, thumb, and sometimes even try to put their whole hand in their mouth at around three months.
Although thumb sucking has a bit of a negative stigma associated with it, typically because parents worry that their baby just won't stop, it is a normal and natural thing for babies to do.
Even the American Academy of Pediatric Dentistry states that sucking on fingers, thumbs, and pacifiers, "is completely normal for babies and young children" and that most children stop before any "harm is done to their teeth or jaws."
Why do so many babies begin to suck their thumb at around two or three months of age? It is at this age that they can more consistently find their fingers and thumb. They are also awake for longer periods of time, giving them more opportunities to suck their thumb, especially when they need to comfort or calm themselves.
So don't worry about your baby's thumb sucking habit. She will likely give it up by the time she is six or seven months old. If not, then she may stop later on, when she is two to four years old. Your child may have some dental problems and may need some help if she doesn't stop by that age though.
American Academy of Pediatric Dentistry. Parent Education Brochures. Thumb, Finger and Pacifier Habits.