A central component of vice presidential influence is allies on the President's staff. Paul Light's most excellent Vice-Presidential Power: Advice and Influence in the White House describes the importance of acquiring allies on the President's staff and of building warm relations between the vice president and president's staff. Mondale excelled at building alliances and placing allies in key positions. Also, because the Carter team was not familiar with the ways of Washington so that Mondale staffers had unprecedented opportunities to fill these gaps.
This has evolved since. While Bush and his team were somewhat suspected by the Reaganites, Bush worked hard to remove this image and - particularly significantly - his close friend Jim Baker was Reagan's chief of staff during the first term. Since Bush had his own team of experienced staffers, Quayle was challenged in building alliances and placing allies. Under Gore, the staffs had very close relationships and Gore was - by all accounts - a key player in the administration. Cheney's record in this regard may not have been as strong as is generally believed. The Bush staff was deeply loyal to Bush and had its suspicions of Cheney and his staffers. Formally, many Cheney staffers also held formal titles on the White House staff - a new development. Titles are important, but not everything. Probably the most important asset of Cheney's influence was (like Mondale's) a sense of what to do and how to do it.
Wall of Biden
In evaluating palace politics, it is important to look at the floorplan.
According to the Washington Post profile National Security Advisor Tom Donilon is very close to Biden. He advised Biden during the Robert Bork confirmation hearings in 1987 and during Biden's 1988 presidential bid. His brother is also a long-time Biden advisor and his wife is chief of staff to Jill Biden.
The new chief of staff, Bill Daley, spent countless hours on the road with Biden during his 1988 presidential campaign.
In effect, that entire wall of the West Wing is occupied by Bidenistas.
Plus, the new press secretary, Jay Carney, had been Biden's press secretary. Off-hand I cannot think of a vice presidential staffer being promoted to the equivalent position on the president's staff. More than half of the West Wing's office space is filled with individuals with close links to Biden. That is unprecedented vice presidential influence!