many times over past couple of years. I95 really is the way to do it and that takes us 14 hours with two drivers switching every 4 - 6 hours ( note 14 hrs if all is going perfectly - weather, traffic, relatively quick gas, food/rest stops ). From where we live it takes us all of 10 minutes to cross into central NJ & pick us I95. Typically if we leave 4:30 a.m. we make HHI by 6:30 p.m.
Have to be honest - - it is a stretch even with two drivers. Well over 800 miles - alot to cover continuosly. Don't know if you're travelling with kids ( ours are older so they handle it well ) but it probably is too much road time in one day for them if they are little ones.
If you have some extra time to spare it really makes more sense to stop overnight on road like Washington DC ( we use Priceline to bid a room at 4 star or higher level you can get a great deal) where you can get see some sites if you arrive early enough. Also can stop in Richmond, VA (we like Chester just south of city-Holiday Inn Express or Marriott) or Rocky Mount NC ( we like Rocky Mount exit where hospital is located less than 5 minutes from I95 - it has nice hotels Hampton Inn, Marriott Residence Inn & Courtyard ) Also, Rocky Mount has a more town/residential feel to it and good places to dine ( Outback Steakhouse, Highway 64 Diner among others nearby ) & even some nice places to shop/browse close by.
Let me mention some family members have taken Garden State Parkway to last exit in Cape May, then Cape May Ferry to Lewes Deleware, then south ( believe it is route 113 but check a map ) through Delewar/Maryland/Virginia penninsula to Chesepeake Bay Bridge/Tunnel into Virginia Beach area and then have reconnected with I95 at some point in that area ( believe you'd take I64) or have travelled a local type route ( sorry number escapes me ) going south into Emporia Virginia or into Rocky Mount NC to reconnect with I95.
Their opinion on doing this was it is great if you really want to take your time getting to Hilton Head and want to avoid I95 as much as possible BUT the stressed this route really isn't do-able in a day because it takes A LONG TIME !
Last suggestion would be ( never done this but considering it because of gas prices !! ) to take Amtrak from NYC into Savannah, Georgia which is virtually Hilton Head Island's next door neighbor ( there are early morning trains that reach Savannah by 9 pm same day ) You can get a rental car then or If you are only travelling to and staying at Hilton Head Island perhaps your resort would even arrange to pick you up at Savannah Amtrak station or suggest a taxi service to call.
Good luck & if we can answer any more questions please let us know.