What is the Old Chinese Pronunciation for the Character 兮?
There is already a good discussion going below the original question. I'll just strip the Wiktionary link and put it down here:

http://zh.wiktionary.org/zh/%E5%85%AE#.E6.B1.89.E8.AF.AD

字   Mandarin  MiddleChinese  OldChinese  Cantonese
兮   xī        hej            /*ɡˁe/      hai4       


And also:

Japanese   Korean   Vietnamese
kei        hyey     hề


So the consonant is along the line of g and h, something down the throat, unlike xi which is in the mouth.

While I don't agree with Phil on "兮 is nothing but a punctuation" and "Dialects were totally unintelligible so people had to write to each other even if they met face-to-face", I agree with him that a standard classic Chinese dialect didn't really exist. There could be many ways to read 兮 in ancient times, especially the vowel, although it's thought to be e, I wouldn't be surprised if it was shifted to a, o or any strange diphthongs in a local dialect.