Any reward for repeating the Nightfall strike?
No, you are rewarded only once. Nightfall only gives rewards at the end of strike. Before you go first time, you can see that it has 'Unknows Rewards' in Director, but has none after completion.
You can still get random engram drops from enemies during the strike, but that's not nightfall specific. Strike bosses do not drop gear in Destiny, but you can get race-specific item from them (e.g. Ether Seeds or House Banner from Sepiks Prime, etc).
So the answer is: yes, you gain absolutely nothing at the end of nightfall strike if you've already done it.