The Wikipedia article on the Gettysburg Address points to several sources which may have inspired Lincoln's phrase; these suggest two different interpretations:
William Herndon, Lincoln's law partner, claimed that Lincoln had been particularly taken by a sentence in a sermon by abolitionist preacher Theodore Parker: “Democracy is direct self-government, over all the people, for all the people, by all the people.” “Over all the people” suggests that of the people signifies that the people are the direct object of this government—a very common construction, as when we speak of the education of children.
Daniel Webster said in a very famous speech, the Second Reply to Hayne, that the Federal government was “made for the people, made by the people, and answerable to the people”, and went on to say “This government, Sir, is the independent offspring of the popular will. It is not the creature of State legislatures; nay, more, if the whole truth must be told, the people brought it into existence, established it ...”. This suggests that of the people signifies that this government arises from the people—an old construction, now little used, but still common in 19th-century literary and rhetorical discourse.
A third possible source, Chief Justice John Marshall in McCulloch vs Maryland, a decision which become a keystone of Constitutional interpretation, marries both these readings: Marshall wrote that "The government of the Union [...] is, emphatically and truly, a government of the people. In form, and in substance, it emanates from them. Its powers are granted by them, and are to be exercised directly on them, and for their benefit."
Herndon is an authority for Lincoln's knowledge of the Parker sentence; and Lincoln, as a lawyer and a student of the Constitution and of the political debate over its interpretation, was undoubtedly familiar with the Webster and Marshall passages. I incline toward understanding the phrase ambivalently, as Marshall uses it: a government of the people is one which emanates from the people and is exercised on them.