Take a look at Thurston's notes (Chapter 13 of his "The geometry and topology of 3-manifolds", available at the MSRI's site: http://library.msri.org/books/gt3m/). Thurston calls these points singular points. (Same as Satake in his original paper "The Gauss-Bonnet Theorem for V-manifolds" from 1957 where he introduces orbifolds, under the name V-manifolds, see http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.jmsj/1261153826) As a geometric topologist, I would use Satake-Thurston's terminology. However, they both work in the geometric topology/Riemannian geometry setting. In the algebro-geometric context you could call them orbifold points to distinguish these points from singular point of the underlying variety.
Google search revealed the following statistics: "stacky point": 328 hits, "orbifold point": 9,750 hits, "singular point" + orbifold: 17,500 hits.
I also like "stucky" and "sticky" points: If you have an isolated singular point $p$ of a Riemannian orbifold, then unparameterized geodesics after entering $p$ get "stuck" there.
On even lighter note, Thurston in his notes explains that the name "orbifold" is the result of a vote he had at his seminar in Princeton. One can have a similar vote on MO to determine what to call points with nontrivial stabilizer.