There isn't a single best month, but ask most guides and they'll pick early season, late October and November. That's when the big bull elephant seals are jousting, which is the signature early sight and essentially finished by the second week of November. The light is at its best, you can be ashore for sunrise and sunset, sites are more accessible before the snowmelt, and it's cheaper and quieter.
South Georgia changes much less month to month than Antarctica does, once you're past that early window.
Roughly how it runs:
Late October and November. Elephant seal jousting, penguins returning, light-mantled albatross courting, birds very active overhead. Macaronis arrive in the first week of November. King chicks are there, as they are all year.
December to February. Stable and consistent. Fur seals dominate the beaches. Most pups arrive from late November through December, so by January the beaches are full of them. Our sources differ on exactly where the peak falls, so treat that as a window rather than a date.
March. Teenage fur seal pups playing, which is a different kind of good.
One practical warning for mid season: territorial fur seal males become a problem from mid to late December, and they're the main cause of cancelled landings there, ahead of weather. Expect maybe one lost landing.
The honest framing is different rather than better. If elephant seal jousting is what you want, you have a narrow window. Everything else is available across the season.