Cornelius is a Latin name of ancient Roman pedigree — stately, long, and impossible to make sound casual. On a dog, it reaches a particular peak of comedic grandeur that owners either find irresistible or completely bewildering. There is no middle ground with Cornelius.
The Ancient Roman Dog
Cornelius was a major Roman gens name, associated with generals and senators. Giving it to a dog collapses centuries of imperial gravity into a single creature who is probably stealing food off the counter. That collapse is the entire joke, and owners of Basset Hounds and English Bulldogs recognize it immediately.
Planet of the Apes
Cornelius is also the name of the sympathetic chimpanzee scientist in the original Planet of the Apes films — a cultured, intellectual character who would appreciate the irony of the name being applied to a dog. Owners who know the reference add another layer of intentionality to the choice.
Nickname Dynamics
The natural nickname is Corny — which is either charming or unfortunate depending on your tolerance for puns. Some owners choose Cornelius specifically to get to Neil or Cornel. See also Archibald and Benedict for the full grand-name experience.