How about the PU Cotsen Children's Library? It's in the Firestone library (Washington and Nassau) and has a wishing well and a cute hideaway tree. Just note that they really do like to keep it quiet! We spent a lot of time strolling around campus. Stop in front of Nassau Hall, visit the gardens behind Prospect House, pop into the free art museum for a few minutes.
Off campus, Thomas Sweet as a PP mentioned, or The Bent Spoon (Palmer Square). PP who mentioned Princeton's good restaurants: huh? Please share some recs!

Not too far away from Princeton is the Grounds for Sculpture.

Sesame Place is only a half hour drive from there. And there are playgrounds. You can go down to the lake and see the ducks. Thomas sweets or Halo Pub for an ice cream treat. And there is a very cool toy store right on Palmer Square now. Honestly though, I grew up very nearby and am struggling to think of things to tell you to do. It is a pretty adult town.

Wow, this is a tough one...Princeton has some great restaurants, and some good cupcake spots for dessert. For a 3-yr old, a pile of dirt could be entertaining. It is a very easy place to walk around - if you walk her, bring a stroller, and he may sleep in no time. Go to the University store and buy her some wacky Tiger gear. But IMO, it is a very adult town. Sorry!

I find I'm going to be in Princeton for a day with my 3-year-old. I need recos for things to do with her apart from the PU Natural History Museum or pretending to be an overachieving mom and demanding a campus tour for her as a prospective student
Thanks for any and all suggestions!