  We introduce a new type of slant curves in almost contact B-metric manifolds,
called $\varphi $-slant curves, by an additional condition which is specific
for these manifolds. In this paper we study $\varphi $-slant null curves in a
class of 3-dimensional normal almost contact B-metric manifolds and prove that
for non-geodesic of them there exists a unique Frenet frame for which the
original parameter is distinguished. We investigate some of $\varphi $-slant
null curves and with respect to the associated B-metric on the manifold and
find relationships between the corresponding Frenet frames and curvatures. We
construct the examined curves in a 3-dimensional Lie group and give their
matrix representation.
