1550s, "cloister, covered walk," from Medieval Latin laubia, lobia "covered walk in a monastery," from a Germanic source (compare Old High German louba "hall, roof;" see lodge (n.)).
Meaning "large entrance hall in a public building" is from 1590s; in reference to the House of Commons from 1630s. Political sense of "those who seek to influence legislation" is attested by 1790s in American English, in reference to the custom of influence-seekers gathering in the large entrance-halls outside legislative chambers.
"seek to influence legislation" (originally by frequenting the lobby of a legislature, to solicit members), 1826, American English, from lobby (n.) in the political sense. Related: Lobbied; lobbying.