Hi,
we have been to Djerba several times with our own car with blue number plates. You need a "Carnet de Passage" for your car which you can get from the Libyan Automobile and Touring Club in Tripoli (sometimes at the border but the office will be closed in the evening and on weekends). For this they need a copy of the logbook, the libyan driving license, passport photos, copy of passport with resident visa, a small fee and a deposit of 500 LYD (fix amount). You can use it for as many border crossings you like. You get the complete deposit money back when you return the "Carnet". First time you cross the border, one sheet of the carnet will be kept by the border authorities and you get a stamp in the carnet. Next time you just need the stamp. Take care that the car is registered in your passport! They check when you come back. For return to Libya, they check if you brought your car with you, it will be registered in a book at the border with a fee of 1 Tunesian Dinar (keep one TDN for this!). You need around 2 hours to get from Tripoli to the border and another 1-2 hours to Djerba and 1 (sometimes 2) hours at the border, most of it waiting in the car queue. People at the border are usually very friendly and helpful, especially to foreigners.
Hope this information helps you.