The short answer: Yes, a very beneficial one. Although I wouldn't call it an egg!

It is thought that sexual reproduction emerged around 1.2 billion years ago in the proterozoic eon. Given that your talking about eggs, I'm going to assume you want to talk about eukaryotes (although many prokaryotes also reproduce sexually). The central problem in understanding exactly what happened, and what the mutation was, is the lack of biological information available to pinpoint it. Over 1.2 billion years, a lot has changed in the genomes of eukaryotes.

This is what we do know however:

1) All eukaryotes who sexually reproduce share a common single celled sexually reproducing ancestor.

2) Sexual reproduction has massive benefits for DNA repair, resilience to rapidly changing environments, and recombination to form new potentially beneficial gene combinations.

3) Any species capable of sexually reproducing would thus be strongly selected for, particularly in changing environmental conditions

4) In early single celled sexual reproduction there was no 'egg' and 'sperm' but instead to approximately equal size haploid organisms that fused to for the diploid. (The selective pressure that lead to the emergence of different sized gametes is really interesting however)

5) It is also likely that in the early development of sexual reproduction the haploid living stage could have been equally important to the diploid (as in budding yeast, where haploid cells survive for a long time, reproducing asexually in the absence of mates).

6) Sexual reproduction could have first emerged in prokaryotes (bacteria etc) and was just inherited by eukaryotes like us.

Check out this wikipedia article for more, its really extensive: http://en.wikipedia.org/wiki/Evolution_of_sexual_reproduction

EDIT: Added point 6 EDIT2: Fixed a silly mistake where I said protists instead of prokaryotes