That's a 6.5 hour flight. They'll probably serve drinks and either a meal or a snack after departure. Figure 2 hours or more before they clear away and turn the lights down. Then they'll wake you about an hour before landing. That gives you less than 4 hours of quiet time overnight. Unless you all can sleep through everything - and even then, you'll have 6 hours sleep at best, you'll be pretty tired when you get to Dublin (cranky kids?). Presumably you'll have luggage with you. Your options are to rent a car (more below) or store your luggage in the city in or near the train station (you'll have to find out if it's possible at whichever station you are departing from and what time they open in the morning). St. Stephen's Green is a pretty park in the center, a few minutes' walk from Trinity College. It opens at 7.30 am (except Sundays, 9.30 am). Most sights and attractions don't open before 10 am.
Car rental - I don't think that's a solution. First and foremost, it's a really bad idea to drive after a longhaul flight. Driving jetlagged and/or sleep-deprived is equivalent to driving under the influence. Add to that, Ireland, like the UK, drives on the left, so that's another distraction/difficulty. Then you may hit traffic getting into Dublin and you'll have to allow enough time to return the rental car and get to the train station. You could hire a car and driver to take you out for scenic views and that sounds like your best option.
