Which is correct - 'everybody has his or her opinion' or 'everybody has their opinion'?
It would be more correct to say "Everybody has his/her opinion."
The use of their in this sentence would be correct, though it is widespread in American speech. Originally, one would say "Everybody has his opinion", because "his" would refer to both men and women. However, as women's rights activists became more vocal, it was deemed improper to use the masculine form to describe men and women.
Their is correct here. The use of their as a singular neuter pronoun has been valid for hundreds of years.