Westley is the name of the farm boy turned Dread Pirate Roberts in The Princess Bride (1987), one of the most beloved films in American cultural memory. Naming a male pet Westley is a statement of allegiance to that film's romantic adventure, delivered with the unusual spelling that distinguishes the character from standard Wesley.
The Princess Bride Connection
"As you wish" remains one of cinema's most quoted lines. The character is heroic, devoted, and has excellent hair. Owners naming pets Westley are usually deliberate about the reference and pleased with themselves, which is warranted. Borzois and Greyhounds, with their romantic-hero bearing, carry the name beautifully.
Spelling as Signal
The WESTLEY spelling is specifically the film character's version. Choosing it over Wesley signals the reference is intentional, not coincidental. The human name Wesley has its own Methodist history, but Westley is firmly the farm boy's name.
Counter-Reading: Core Audience Is Aging
The Princess Bride's most devoted audience is Gen X and older Millennials. Younger owners may not carry the reference with the same intensity. The name still works without the film. It just means more with it. And you will be explaining "As you wish" to someone at the dog park within the first month.