Wordorigins.org

To lobby, meaning to influence legislators, and lobbyist, meaning the people who do so, are references to advocates who frequented the lobbies of the US state capitols and eventually the US capitol in Washington, DC in an attempt to button-hole legislators. The verb has been in use since at least 1837. This quote is from the 6 October Cleveland Herald of that year:

Gen. Bronson...spent a considerable portion of the last winter in Columbus, lobbying to procure the establishment of a Bank at Ohio City.

The noun lobbyist has been around since at least 1849. From the 31 July New Hampshire Gazette:

This interest and this feeling were taken advantage of and subjected to a constant stimulation by a score of indefatigable lobbyists, who kept up an untiring attack upon the members.

Also common in the 19th century were the forms lobbier and lobbyer. From James S. Buckingham’s 1841 America:

In one of those instances, the parties employing a Mr. Hillyer, of New York, as a “lobbier,” to promote the passing of a bill through the Trenton legislature...refused to pay him.

It is commonly, but incorrectly, asserted that the terms originated as references to the lobby of the Willard Hotel in Washington during the administration of President Grant (1869-77). Grant supposedly frequented the hotel and he would refer to those who approached him there seeking favors as lobbyists. Not only was the term in use long before Grant’s presidency, but the Willard Hotel, which opened in 1850, did not even exist when the words were coined.

(Source: Oxford English Dictionary, 2nd Edition; Mathew’s Dictionary of Americanisms; NewspaperArchive.com; ADS-L)