Here are 20 unique cat breeds you might want to make your character. Keep in mind that wild cats are very unlikely to be purebred, so if you do make your OC one of these breeds, explain why in their backstory somehow. For example, they could be an escaped Kittypet.

American Curl
Appearance: curly ears; long or short hair; different colors/patterns
Temperament: friendly; outgoing
Picture:

Balinese
Appearance: long haired; color point fur; blue eyes
Temperament: Friendly; playful; talkative
Other details: have "traditional" and "modern" styles that look quite different
Picture:

Bobtail
Appearance: bobbed tail; any color/pattern; short or long hair; might have tufted toes
Temperament: Extroverted; playful
Other details: American Bobtail and Japanese Bobtail are different breeds
Picture: I can't find a good pic, but here's a pic of a stray cat by my house (not sure what breed?) who had a bobbed or missing tail

Chausie
Appearance: Medium or large; meant to be black or tabby, but are sometimes other colors/patterns; sometimes "grizzled" tabby which is only found in Chausie cats
Temperament: Energetic; smart
Other details: hybrid breed with jungle cats; good at jumping and running
Picture:

Devon Rex
Appearance: Short, wavy fur; big ears
Temperament: Energetic; friendly
Other details: great at jumping
Picture:

Maine Coon
Appearance: very large; long haired; any color/pattern; tufted ears
Temperament: Friendly; smart; like water; talkative
Other details: Take years to get fully grown; water resistant fur; adapted to survive in cold, snowy places
Picture:

Munchkin
Appearance: Very short; any color of pattern; can be short or long haired
Temperament: Kind; outgoing; smart
Other details: Controversial breed due to the fact that they are bred to have deformities; might have mobility issues
Picture:

Napoleon / Minuet
Appearance: long or short hair; short legs
Temperament: friendly; active
Other details: Cross between Persian and munchkin; probably controversial like the munchkin
Picture: (Honestly not sure if this is a minuet or regular Persian; it's the best copyright-free image I could find)

Norwegian Forest Cat
Appearance: long hair; large; any color eyes
Temperament: smart; friendly
Other details: Natural breed; good climbers; do well in the cold; might have health issues
Picture:

Ocicat
Appearance: short spotted fur; large; variety of colors
Temperament: friendly; smart
Other details: resemble an ocelot, but aren't ocelot hybrids
Picture:

Oriental Shorthair
Appearance: Giant ears; short fur; skinny; variety of colors; usually brown eyes
Temperament: quiet, social, smart; not playful as adults
Other details: athletic

Persian
Appearance: Flat muzzle; long haired; any color/pattern
Temperament: quiet; friendly
Other details: Prone to polycystic kidney disease
Picture:

Savannah Cat
Appearance: usually spotted fur; variety of colors, sometimes diluted; tall
Temperament: loyal; like water
Other details: Cross between a serval and domestic cat; good jumping
Picture:

Scottish Fold
Appearance: Folded ears; long or short hair; any color or pattern
Temperament: friendly; playful; dislike being alone; stubborn
Other details: May have health issues
Picture:

Selkirk Rex
Appearance: Long or shorthair; large; any color
Temperament: friendly; playful; laid back
Picture:

Siamese
Appearance: Blue eyes; color point fur; large ears
Temperament: Very talkative; friendly; curious
Other details: crossed eyes used to be common
Picture:

Singapura
Appearance: Small; big eyes; short fur; ticked tabby fur; only comes in one color
Temperament: Active; friendly
Picture:

Somali Cat
Appearance: Long haired; ticked fur; variety of colors
Temperament: Smart; friendly
Other details: often called "fox cat"; might have health problems; teeth are brushed as pets
Picture:

Sphynx
Appearance: Hairless; come in different colors; may or may not have whiskers; large ears
Temperament: Extroverted; energetic; smart
Other details: Sensitive to cold weather; can get sunburnt; need regular baths
Picture:

Toyger
Appearance: Striped like a tiger; short fur
Temperament: active; friendly
Other details: name meaning "toy tiger"
Picture:

SOURCES
• pictures are copyright free images
• https://en.m.wikipedia.org/wiki/Sphynx_cat
• https://en.m.wikipedia.org/wiki/Maine_Coon
• http://mentalfloss.com/article/80011/7-short-facts-about-munchkin-cats
• https://en.m.wikipedia.org/wiki/Munchkin_cat
• https://en.m.wikipedia.org/wiki/Scottish_Fold
• https://en.m.wikipedia.org/wiki/American_Bobtail
• https://en.m.wikipedia.org/wiki/Oriental_Shorthair
• https://en.m.wikipedia.org/wiki/American_Curl
• https://www.kisspng.com/png-american-curl-american-shorthair-american-bobtail-1834817/preview.html
• http://cfa.org/breeds/breedsab/americancurl.aspx
• https://en.m.wikipedia.org/wiki/Persian_cat
• https://en.m.wikipedia.org/wiki/Savannah_cat
• https://en.m.wikipedia.org/wiki/Toyger
• https://catappy.com/facts-about-toyger-cats
• https://en.m.wikipedia.org/wiki/Devon_Rex
• https://en.m.wikipedia.org/wiki/Selkirk_Rex
• https://en.m.wikipedia.org/wiki/Somali_cat
• https://cattime.com/cat-breeds/somali-cats#/slide/1
• https://en.m.wikipedia.org/wiki/Minuet_cat
• https://www.petguide.com/breeds/cat/minuet-cat/
• https://en.m.wikipedia.org/wiki/Chausie
• https://en.m.wikipedia.org/wiki/Ocicat
• https://en.m.wikipedia.org/wiki/Balinese_cat
• https://en.m.wikipedia.org/wiki/Singapura_cat
• https://en.m.wikipedia.org/wiki/Siamese_cat
• https://cattime.com/cat-breeds/siamese-cats#/slide/1
• https://en.m.wikipedia.org/wiki/Norwegian_Forest_cat