I think if you just walk down the Highline then proceed directly southward (towards the tip of lower Manhattan) and get a little lost wandering the streets of the West Village aiming for Greenwich and Christopher that you’ll find lots of quaint streets and interesting shops on the busier streets. Turn left on Christopher until Bedford then go left on Bedford until Carmine or Downing and cross 6th (Ave of fhe Americas) and head over to Washington Sq Park and NYU (actually 90% of the Village is NYU now (joke) but you’ll still see lots of cool buildings.
You’ll probably want to have a map/phone with you as this area of the West Village is the main exception to the grid plan on Manhattan. All kinds of interesting things to see in this zone that is much more peaceful than Midtown madness.
I absolutely agree that walking the High Line and stopping at the Whitney and Chelsea Market is a lovely way to spend time in NYC.
Rather than Ellis Island, I prefer Governor’s Island for that sort of island getaway by boat. But I’ve only been on weekends and don’t know if all options are open on weekdays.
I’d 100% suggest Sleep No More (https://mckittrickhotel.com/). It is an exciting and unique immersive theater experience that is ending soon. There’s really nothing else like it in the USA. Would recommend skimming the wiki for Macbeth as a refresher. Pre-show, I’d highly recommend dinner at Shukette or Cookshop which are walking distance (make reservations).
West Village is classic. You have Angel’s Share as a world-class cocktail bar, Nami Nori is a casual but elevated handroll spot, a fave of mine. Two Boots Pizza is great for a slice and is funky. Honestly worth a shot taking her to Smalls Jazz Club and just seeing if it connects with her - it’s small, intimate, and very lively.
Lower East Side has Mercury Lounge which is a great non-Jazz music venue and there are tooooons of bars and restaurants in the area. Scarr’s Pizza for a slice is life (@Mike_Cohen is right - old location was best). You could dom a whole food walking tour from Chinatown to Little Italy. Thai Diner is one-of-a-kind.
Upper East Side… Temple of Dendur at MOMA is special, in my opinion. Guggenheim is great and pretty accessible as far as museums go. Walk to Earl’s Beer and Cheese after.
Other recs in the thread are great. Enjoy!
I worked in the Chelsea Market Bldg in 2001/2002 just before things really changed in the area. Not sure if anyone here remember what is was like back then. Almost hard to believe how quiet it was in those days.
btw, that building was once a Nabisco factory.
pretty simple, just download the app. same price as a subway ride just more scenic. could go all the way out to the rockaways if you like. midtown east to williamsburg would make a fun afternoon/evening.
the only current reason to go inside chelsea market is for Amadou Ly’s profound pastries at Alf Bakery and right next door is Dickson’s for a hot dog or great pork and other meats. The rest is a complete shit show.
The Dickson’s hot dog is delicious. But it tends to stay with you. Or at least it does for me.
I hear you. I guess there are some decent places to eat there like Tacos but now NYC has plenty of good Mexican options without the insane crowds of Chelsea Mkt. It used to by one of my sanctuaries to find some peace and now it is pure mayhem. Oh how nice it was to step into the huge refrigerated fruit and vegetable store in the summer and get some fresh squeezed juice. 20 visitors on the entire ground floor of the building was a crowded moment.
amadou is the man! rip arcade tho 
I always love reading these threads to see what others recommend. You are not going to be in NYC long so hard to recommend a lot things due to time constraints. The best part of NYC is just walking and exploring. If I was taking a 21 year old around most of the spots I would go to would be in Brooklyn.
Given your time constraints I will suggest some cool things that are close together. The Dime Square area is an excellent idea. All of the cool kids hang at Colbo. The owners are two young, very stylish Israel kids who have really cultivated a nice community. Scarrs is right down the street so you could wait in line while your daughter shops. Personally I don’t think Scarrs is worth the wait. L’Industrie in the west village is the best clice in Manhattan. Skin Contact the natural wine bar run by Eben of Chambers is also very close. Don’t miss Colbo for your daughter’s sake.
You can spend an afternoon walking around this area going in and out of shops and there a lots of small galleries. Then go to any of the many restaurants for dinner. Cervos is excellent but you either need a reservation or get there 10-15 minutes before opening. If you can’t get into Cervos, Tolo (Chinese spot run by Parcelle) is a good alternative.
After dinner go to either the Swan Bar at 9 Orchard or any of the Natural Wine Bars where the young people hang, your daughter will love them:
Another great dinner option is Gem Wine Bar which is close by, a very stylish 20ish crowd of mostly young women as the Chef has quite a following.
If you do head over to the East Village Tokio 7 is one of the best resale / vintage shops.
If you happen to be really adventurous I would suggest a trip to Public Records, the best music venue in the United States, a late night dinner at the bar at The Four Horsement followed by a night cap at Nighmoves their music spot next store!
Thanks to everyone for the suggestions. It blew my family’s mind when I told them my post generated so many responses!
Please report back! And fire away any ad hoc / emergency questions.
I will add in a visit to the Lobster Place for some oysters in Chelsea Market
Suzanne,
I agree. The Lobster Place is a shadow of it’s former self as a fish market, but if you want a snack of fresh shucked oysters, they have a very good selection. It is a madhouse though.
Interesting the hear how it has changed. When I was working in the building there was never even the slightest line at the lobster place or anywhere else in the building. Used to eat lots of ice cream from the creamery. Looks like that is gone now and replaced with gelato.
Robert,
I’m pretty sure I’ve been to Colbo with my wife and kids. You can get custom painted sneakers upstairs? I thought the whole place was weird (but I’m 54) and I didn’t understand who was buying stuff from there, but my wife and kids (22 and 20) seemed to think it was pretty cool and happily browsed around.
This is a bit of a tangent, but my youngest is a fashion design major and an artist. I wish she would put her stuff out in the world more…based on the custom painted clothing I saw at Colbo and the prices they were asking (I don’t know if people are actually buying at those prices) I think my kid could make some real money. This is obviously coming from a biased father.
Hey Mike - I don’t think they have an upstairs. And honestly it is not weird to me, if any thing I think all of the tan and muted colors that the 20 and 30 year old set like that they specialize in is a bit tame for me. And I am older than you 
I do know it is a nice scene. I know Tal from the music world so I always stop in for a super strong expresso and buy a few records to support them.
Ahhh…I may be thinking of a different place as I remember there being an upstairs. Whatever I’m thinking of is on Orchard, 2-3 blocks north of canal on the east side of the street.
I’ll have to check out Colbo the next time we are down there.
I think I know the place you are thinking of, its a block or so from Colbo.
I’m very weather dependent.
My favorite museum is the Met. There is always a less crowded part.
If the weather is great, outdoors is my choice.
Love the high line.
My favorite walk starts downtown (at battery park or city hall for a little shorter walk), then after City Hall, little Italy, Chinatown, Lower East side.
Wind up at Houston Street where there is a trio of foodie destinations- Katz’s , Russ and Daughters, and Il Laboritorio del Gelato - the best pastrami, smoked salmon and gelato in the US , IMO, within 3 blocks of each other.
On the way, you can hear traditional Chinese musicians in a park (Columbus Park I think), and the tenement museum is great. Just a ton of activity and the neighborhood changes every 6 blocks. If you can’t make it yo Houston without stopping for good, plenty of great stops. Parm has great chicken/eggplant parm and meatball sandwiches. Probably good wine bars on the way too,