Hey 007 lived in nyc for 7 years lived in Brooklyn for two of them, (loved Brooklyn BTW) did a uhaul move back to the cuse last year and did the SYR to BK run often, 81/380/80 is the way to go. When you are about 20 minutes into Jersey across the PA border pop on AM1010 WINS on your radio for traffic every 10 mins on the 1's, by the time you get to the turn offs you will know the shortest wait time at the bridges and tunnels and turn off accordingly. The Holland is the most challenging to get to from the Syracuse angle and the Pulaski skyway is closed but it is an option. I often took the Lincoln or the George Washington and took the west side highway downtown (if not backed up) because the Holland is often jammed the worst. If the west side highway is clear and Holland is backed up 10mins more than the Lincoln take the Lincoln and then if the Lincoln is 10 min more than the George take the George because each one you are going 10mins more out of your way to avoid traffic. Best time ive done is 4.5Hrs to central BK via the holland, 5.5 or 6 is not out of the question with the uhaul. I agree with above posters choose off of rush hours definately not Monday night this weekend that will shorten the trip.
Which part of BK is your daughter living as that will impact what route bridge/tunnel you should take once you are in Manhattan? When you park the truck check the parking signs immediately traffic cops are crawling all over the city put your flashers on and double park and alternate trips so someone is always with the truck if you have to the cops do understand moving is crazy down there.
Mindset is important, Know that you will get caught in traffic somewhere in this process don't be in a rush to get around people, be fluid with your routes if you have to go around the block to make the correct turn so be it and stay assertive people will try to block you out so they can get someplace 30sec sooner.
Holland tunnel is restricted to class 3 trucks (3axle) or less your uhaul should be fine, I think they call them all "Vans" but check with the uhaul people to make sure the class. Just didn't want you to freak when you saw the no trucks symbol.
You likely can not take the uhaul on the garden state parkway or the FDR Drive/East River Drive or Onondaga lake parkway for that matter. Your car EZ pass does not usually work for the uhaul different tolls depending on size check that at uhaul when you pick up.
PM me if you need to I can give you my number.