There's been a lot of talk in recent years of eliminating the Electoral College, or making all political campaigns entirely funded by the public, among others. But those would likely require constitutional amendments if they were to pass judicial scrutiny.
Knowing how polarized Congress and this country is, what political reforms could we pass without needing a constitutional amendment? Here are some possibilities:
1. campaign financing transparency
2. nonpartisan congressional redistricting
3. further erode the filibuster
4. automatic voter registration/updating
5. mandate that the options of early voting and vote-by-mail be available in every state
1. Probably to some extent, but every attempt by congress to regulate campaign finance has tended to make things worse. We probably need a constitutional amendment to really clean up the Super PAC mess.
2. This is theoretically possible. Congress has already mandated single-member districts by law. The voting rights act regulated racial gerrymandering. Additionally, the Supreme Court has held that excessive partisan gerrymandering might be unconstitutional but the Court was divided over whether a judicially manageable solution exists. That leaves the door open for congress to regulate the problem. That being said, I don't know what the enforcement mechanism would be. Congress might be able to mandate "no partisan redistricting," but almost certainly can't actually require a state to create a non-partisan state agency to draw the districts.
3. Sure, you just need a Senate majority leader willing to do it and a majority of senators in support of the rule change.
4, 5. Article I, Sec. 4 gives congress the power to regulate the "The Times, Places and Manner of holding Elections for Senators and Representatives," congress could probably mandate early voting and voting by mail. However, congress probably could not broadly regulate state voter registration or eligibility to vote in congressional elections. The Constitution states that the eligibility requirements for voting in a U.S. congressional election are the same as the requirements for voting for "the most numerous Branch of the State Legislature."