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

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.

There are many lasers out there that can remove your mole including q switched lasers anywhere from 650nm to 1200nm. Intense pulse light is another possible way but with less evidence and experience out there. Excising the mole is the most assured way to fully remove it but with the cost of a visible incision. If done right you can make that incision look super however. One thing to think about if there is any recent change to the mole you should consider getting it looked at by a pathologist and hence biopsied.
Consulting a Board Certified Facial Plastic Surgeon would be something that I recommend. We specialize in the face and are highly qualified to help you with this.

The location may be a tip off that you may not have a mole ( this is a lay term...but we dermatologists have adapted it to generally mean a pigmented lesion..the proper name for it being a melanocytic nevus in that there can be nevi which are not melanocytic such as a connective tissue nevus).
Now that I have cleared up the nomenclature: you may have a small growth called a papilloma. This is a benign excrescence, often due to rubbing the eyes. This is a very common growth around the eyes.Removal of this is simple with an appropriately named iris scissors.
Otherwise, I would recommend a biopsy before excision. Although chances are slim, but what might appear to be a mole, might actually be something more ominous. The less likely, but more serious tumors, would include a basal cell carcinoma, a sebaceous gland carcinoma, or a melanoma.
Once the lesion is biopsied, whether it should be removed by laser or a simple surgical excision would be up to you and your dermatologist or plastic surgeon. Surgical excision of course would be much less expensive, but your judgement should not be clouded by this sole consideration.