U.S. citizens have tried to get national initiative power since 1907, when Rep. Elmer Fulton (D,OK) introduced House Resolution 44, which failed.
The next official attempt was the National Voter Initiative, Senate Joint Resolution 67, introduced in 1977 by Senators James Abourezk (D,SD)and Mark Hatfield (R,OR). Unlike Rep. Fulton's proposal, SR 67 would not have enabled Constitutional amendments by ballot initiative. Also introduced at this time was House Joint resolution 544, whose prime sponsor was Rep Guy Vander Jagt (R,MI), which also lacked amendatory initiative power.
The 1977 attempts were made in the aftermath of the unpopular Vietnam war, the resignation of President Richard Nixon, and various Congressional scandals. Both the 1907 and 1977 attempts failed to get Congress to share power with the people.
Since that time, another supporter of the National Voter Initiative, Sen. Mike Gravel (D,AK) discovered that citizens do not need to beg Congress to share power. The Founders of the U.S. faced the same problem: the existing 13 Legislatures refused to share real power with the nascent USA. The Founders pondered and instead had The People ratify the Constitution at the Constitutional Conventions. James Madison said "The people were in fact, the fountain of all power, and by resorting to them, all difficulties were got over. They could alter constitutions as they pleased. It was a principle in the Bills of rights, that first principles might be resorted to." (He refers to Amendments IX and X. This is his 2nd response in the 1787 Debate)
By the same reasoning that "the People are Sovereign", Senator Mike Gravel and company have enabled registered U.S. voters to vote to ratify the National Initiative for Democracy, which can be done at Vote.org since 2003. The right to do this has been vetted by leading Constitutional scholars, including Yale's Akhil Reed Amar in his paper Popular Sovereignty and Constitutional Amendment.
The National Initiative For Democracy is the most ambitious such proposal, allowing amendments as well as statutes by initiative. It gives citizens the deliberative process legislators have, including public hearings, expert testimony, and reports about proposed initiatives, to be conducted by randomly-selected Deliberative Committees, and disseminated by all media.