Grady registers 62 times at rank 1647 on male pets. It's an Irish surname that became a given name, with the same warm, slightly rough-edged quality as Clancy or Brady. On a dog, it reads as reliably good company: the kind of name that sounds like the dog will always be happy to see you but won't be too earnest about it.
The Irish-American Surname Tradition
Grady comes from the Irish O'Grádaigh and has been used as a first name in Irish-American communities for generations. It sits in the same register as Brady and Clancy: friendly, slightly roguish, comfortable in its own skin. Supporting characters who are reliable rather than glamorous carry this name in American pop culture, which is exactly the brief for most family dogs. The human-name context is at /names/grady.
Sound and Breed Fit
GRAY-dee is two syllables, open and warm. The name calls well and has a natural friendliness in the sound. It suits mid-size male dogs with sociable, confident personalities: Labs, Boxers, and mixed breeds with an Irish Setter quality to their personality.
The Counter-Reading
Grady is not trying to be interesting. It's trying to be dependable, and it succeeds. Owners who want something that makes people ask "where did you get that?" should look elsewhere. Owners who want a name that just fits, every day, will find Grady reliably satisfying.