Inexpensive, Low-risk Method of Removing a Mole Near the Eye?

The easiest way is to have a shave biopsy done. The mole is injected with lidocaine and then shaved off like a sliver with a scalpel. The base of the wound is cauterized with a bovie (hot rod like instrument). The removed mole is sent for biopsy. 
Regards

Least expensive is shave biopsy, often with good cosmetic results.  An advantage of this method is that you get pathology to review your mole to be sure it is not malignant (this is always a good idea) and that there are no stitches to deal with.  It heals like a scratch heals.  Also, there is no need to be concerned about general anesthesia costs/risks when you choose the shave biopsy technique.