It may depend on how you want to define "nicest".
For ocean swimming, May - October would be "nicest". To avoid crowds, September - February would be "nicest". To enjoy professional golf tournaments, April would be nicest (the Heritage Classic starts this Thursday). For warm weather, May - October would be "nicest". For cool weather, December - February would be "nicest".
We have enjoyed HH at all times of the year and every season has something to offer. For all around pleasant temps, minimal crowds and maximum available activities, March - May and September - mid November are our favorite times to be here - but we also really enjoy early morning walks on the beach followed by an ocean swim (water temps in the mod-80s) in June, July, August and September. Even December - February, while the weather is a bit more unpredictable, are great because the crowds are low, many restaurants offer specials and it's a great time for side trips into Savannah, Charleston or just exploring Hilton Head itself.
I think that January - December are "nicest".