I'm not sure if this will help or not, but here goes...
I had stayed at the Millennium Hilton right near the UN several years ago. I picked it because it was near a wedding I was attending, but it was also a bit cheaper than the places right on or just off of Times Square or Broadway. I also found the rooms to be larger than what I'd expect for Time Square and Broadway. I had an incredible view and it was also pretty quiet. Don't get me wrong...there are times when I love the buzz of the city, but it was nice to have things a bit more peaceful. And with the UN right across the street, security is really great in the area. That being said, Lily may want the iconic, middle of it all hotel.
If you want more luxury, I'm also a fan of the Pierre. It's right across from Central Park (lower portion) on the corner of 5th and E 61st. My university has a satellite building just down the street, so we had a few occasions growing up to pop in there.
I grew up in NY metro New Jersey, so most of my experiences are taking the train or bus into the city. Whenever I go back up, the museums are my must do, with The Met being my favorite (Natural History is second). It's way up town, but The Cloisters is also a nice stop and I believe it's still affiliated with The Met. If weather is good and you all like animals, The Bronx Zoo is usually a hit. This may be the Jersey girl in me speaking, but when going to the Statue of Liberty, I prefer to access it from Liberty State Park on the NJ side...same goes for Ellis Island. Hey, you'll have your car and it would be an excuse to drive through one of the famous tunnels. But do all of the touristy stuff...whether it be a bus tour, Circle Line, a building tour, etc. If there's a Broadway Show that's a must, start looking into tickets. If you can play it by ear, check out a TKTS tickets booth to see what you can get on discount. Parking is not exactly cheap in the city. I only have experience parking as a day guest, but whatever hotel you select may have some suggestions on nearby garages that are good. Subways are the way to go from a cost perspective, but when visiting as a tourist, I like to be above ground. That's not saying take a cab or walk everywhere, but walking is a great way to see the city. Also, check to see if there are any shows that you like that film live in the city. Each has their own hoops and such to go through to get tickets, but it's a pretty neat experience to be in the audience for a live show. I had a total geek out moment in 2002 when I got us tickets to Letterman.
