Answer:
A constellation in modern astronomical parlance is defined by a region of the sky. It contains ALL the stars in that region, including the billions located in distant galaxies, so an exhaustive list is impossible.
The ones with "names" (as opposed to designations only) are:
alpha Cancri - Acubens
beta Cancri - Al Tarf
gamma Cancri - Asellus Borealis
delta Cancri - Asellus Australis
epsilon Cancri - see below
zeta Cancri - Tegmine
lambda Cancri - Kwan Kei
xi Cancri - Nahn
The story of epsilon Cancri is a bit compllicated. Originally that designation referred to an open cluster of around 1000 stars rather than a single star. Currently that cluster is called Praesepe, and one of the brighter stars within that cluster is e Can. However, as far as I know the current star designated e Can does not have a "name"; Praesepe (aka M44 or "the Beehive Cluster") is the name for the cluster as a whole.
To make matters even more confusing, while Praesepe is also called the Beehive Cluster, Praesepe actually means "manger."