The dog’s name was Nipper:

In 1898, three years after Nipper’s death, Francis painted a picture based on a photograph of Nipper listening intently to a wind-up Edison-Bell cylinder phonograph, substituting a disc gramophone for the phonograph. On February 11, 1899, Francis filed an application for copyright of his picture “Dog Looking At and Listening to a Phonograph.” Thinking the Edison-Bell Company might find it useful, he presented it to James E. Hough who, in a move that would eventually result in Edison exiting the record business altogether, promptly said, “Dogs don’t listen to phonographs.” On May 31, 1899, Francis went to the Maiden Lane offices of The Gramophone Company with the intention of borrowing a brass horn to replace the original black horn on the painting. Manager, William Barry Owen suggested that if the artist replaced the entire machine with a Berliner disc gramophone, the Company would buy the painting. A modified form of the painting became the successful trademark of Victor and HMV records, HMV music stores, and RCA. The trademark itself was registered by Berliner on July 10, 1900. (wikipedia)

More info about Nipper over at DesignBoom.