  We consider the family of hyperelliptic curves over $\Q$ of fixed genus along
with a marked rational Weierstrass point and a marked rational non-Weierstrass
point. When these curves are ordered by height, we prove that the average
Mordell-Weil rank of their Jacobians is bounded above by 5/2. We prove this by
showing that the average rank of the 2-Selmer groups is bounded above by 6. We
also prove that the average size of the $\phi$-Selmer groups of a family of
isogenies associated to this family is exactly 2.
