For those early morning flights I always choose the courtyard logan. For little more than the price of parking at Logan, you can stay in a nice room and park your car for up to 10 days. There is a good casual restaurant, free airport shuttle and nice lobby. Loads of pilots and flight attendants stay here and you know they know where to stay! Recommended
Oh by the way, there was some confusion getting picked up from the airport to get my car. They were accommodating and reimbursed me when I took a cab.. Great Service.