Clearly very little interest in this question!
So, rather than flog a dead horse (or donkey or elephant!), I'll pass the baton to Zag / Andy...
The answer was:
The elephant's link to the Republican Party was also down to Thomas Nast, who's considered the father of the modern political cartoon. He first used it in an 1874 Harper's Weekly cartoon. Titled “The Third-Term Panic,†Nast's drawing mocked the New York Herald, which had been critical of President Ulysses Grant's rumored bid for a third term, and portrayed various interest groups as animals, including an elephant labeled “the Republican vote,†which was shown standing at the edge of a pit. Nast employed the elephant to represent Republicans in additional cartoons during the 1870s, and by 1880 other cartoonists were using the creature to symbolize the party.