So I have been tasked by a relative to setup his .17 Varmint Express with an optic. I will be purchasing a Vortex Crossfire II 2-7x32 unless there's a better, cheaper (the Vortex can be had for just over $100) option for him. What rings (and possibly base?) do I need to make that setup work?

I'm about as unfamiliar with lever actions as you can get.  

Burris has a good rimfire in 3x9.  Are you mounting a pic rail or using the dovetail?

I'll take a look at the Burris. I'm open to using either a pic rail or the dovetail. Is there an advantage to using one over the other?

Some people don’t like a pic rail on a lever.   Dip makes a rail for Henry rifles.  Tallly makes great dovetail rings that will not ned the pic rail.   I had to go through this with my daughter and her lever action.