If you want to skip the sight-seeing and stereotypically touristy things to do in NYC, I suggest doing some major shopping. One place I would highly recommend for jewelry is the jewelery/earring district located at around 31st-32nd and Broadway. There's basically rows and rows of stores selling really inexpensive jewelery.
A quick photo of the inside of a wholesale jewelry store
While one would expect that with all of this selection I would be buying loads of things, I am (fortunately?) very picky about jewelery. Also, I had to make sure everything fit in my suitcase (sadly). Here's my two favorite pieces I got from shopping in the Jewelry District:
Awesome statement necklace
I love the color of this bracelet
