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

The least expensive method to removal a mole near the eye is probably a shave excision.  I would recommend that you have the lesion sent to pathology to confirm exactly what it is. 

First of all, it would depend on the cause of the mole. The word mole is generic and could denote a benign or malignant lesion. Most benign lesions close to the eye,can be shaved off, cauterized, frozen excised or lasered. If in doubt the mole should be biopsied.

The most important factor is finding a surgeon with experience in the face and he/she will suggest the best method of removing a mole, as the characteristics of each mole are different.  I would focus on what is the best method to both remove the mole, as well as leave as imperceptible scar as possible.  Their are multiple techniques for removing a mole including a modified shave technique as well as direct excision and closure.

Most lesions around the eyes are skin tags or warts which only require a 2 second snip for about $150 and is painless. True moles can be excised under local anesthesia which takes only a couple of minutes for about $250-350. 

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