Let $ (M,g) $ be a 2D Riemannian manifold with boundary. The boundary normal coordinates $\psi $ are constructed roughly as follows: in a sufficiently small neighborhood $ U $ of $ \partial M $, for each $ p \in U $ there is a unique $ q_p \in \partial M $ minimizing the geodesic distance to the boundary, ie. $ d(p,q_p) = d(p,\partial M) $. Fix some coordinate system $ z: \partial M \rightarrow \mathbb{R} $, and then let $ \psi(p) = ( d(p,q_p), z(q_p) ) $.
Can some one point me to a reference where this is carefully defined and made precise?
If memory serves, Bishop and Crittenden's book Geometry of Manifolds has exactly what you want.