Taking Zamp's ideas and developing them. We have some great risks of stereotyping here, but here would be my thoughts:
Hinduism - +

-

(Caste and vegetarians)
Hinduism is known for its pluralism.
So I suggest that influence with city states is greatly increased (like Philanthropy and/or Greek UA)
Buddhism - +

(Nirvana)
Buddhism is known for encouraging peaceful states of being and building huge monuments!
So I would suggest zero

for number of cities and +20%

for wonders (like a part combination India/Egypt UA)
Judaism - +

(Helping neighbors)
Judaism's greatest modern contribution may be through the arts, journalism and finance.
So I would suggest +5%

increasing by +5% for each later era with better communication, and +10%

, both empire-wide.
Christianity - +

+

(Paying the Papacy and art)
Christianity is an evangelical religion with a strong work ethic.
So I would suggest +0.5

/city population point in each city and +15% troop bonus.
Islam - +

+

+

(Trade, emphasis on learning, welfare)
Islam is an evangelical religion with a trading and science contribution.
So I would suggest +10%

and +10%

empire-wide, and +15% troop bonus.
Confucianism - ++

(Huge emphasis on learning)
Confucianism is known for the importance it puts on social structure.
So I would suggest new social policies should be cheaper to get by 33%. (like Free Speech/Redentor)
Secularism - ++

+

(Mix of cultures, accepting)
Atheism - +++

+

Scientist (Science science and more science)
With no adopted religion, there is more freedom for original thought.
All science buildings (libraries etc.) are more effective with all normal +50% bonuses rising to +75%, plus +25%

Scientist points.
Taoism (missed by Zamp) could emphasize the acceptance of and working with natural flow.
So I suggest -50%

for number of citizens, plus units have no terrain penalty in friendly territory.