I'll add my vote for a Browning BLR in .243. The "lightweight" version of the rifle has an aluminum receiver, it light, fast to point, accurate, and it comes with the receiver pre-drilled and tapped for scope mounting.

The other lever-action rifle I would recommend isn't available new, it would be a Savage 99 in either .22-250 or .243. The benefit of the Browning is that it has a removable magazine box, and the Savage has an internal rotary magazine. Both are accurate rifles, fast-handling. The Savage 99's are starting to move up in value as collectors discover them. 

The Henry Big Boys you pointed at aren't really varmint rifles, they're all chambered for straight-walled handgun cartridges. But in the Henry Big Boy rifles, the cheapest ammo would likely be feeding .38 Specials into a .357/.38 rifle.