What could cause my hair to stop growing on the right side of my head?

This needs an in-person evaluation. I suggest seeing a hair loss expert now to evaluate and go over your condition and provide you information about your best treatment options. There are great non-invasive options like prp/progesterone and/or hair transplantation. See an expert. Best, Dr. Emer

I suggest a consultation with a cosmetic dermatologist with experience in hair restoration and maintenance. Some great options for hair restoration and maintenance are prp injections which can stimulate hair growth. Hope this helps.

For
many, hair loss is hereditary. Dihydrotestosterone, or DHT, is a
derivative of testosterone and is what has been found to cause hair loss
in both men and women. About 50% of men and 20% of women have this
defect one one of our hereditary genes which overproduce this. As a
result, high levels of DHT leads to problems such as hair loss and
balding.
However, there's also other causes of hair loss such as stress, diet, trauma, and some medications.
The number one first step I recommend is a
comprehensive consultation with a hair regrowth specialist who will
perform lab tests to determine the underlying cause of your thinning.
Without labs, no-one can really know what your true cause is and any
treatment will be temporary at best. From here, you can receive the best
treatment for your case. Good luck to you and do not hesitate to reach
out with more questions.

One would need to see the scalp up close. There are actually a lot of diagnoses to consider and they can only be properly evaluated by examining the scalp. Age related hair loss, androgenetic alopecia, scarring alopecia, alopecia areata, traction, inflammatory localized dermatoses all need to be considered. Be sure to see a dermatologist. 

It is not usual to one side of the hair to stop growing. I would have to examine you first to see what your hair looks like all over the scalp