"In 1976, Jimmy Carter considered Mr. Rodino as a running mate. ''Tippy [Tip O'Neill] called me all excited, saying I was at the top of the list,'' Mr. Rodino said. ''I said, 'No, I'm not. I like what I'm doing and you need me where I am, as chairman of Judiciary.' Tippy was surprised. So was Carter.'"
https://www.nytimes.com/1989/01/27/nyregion/after-40-years-making-the-law-rodino-now-teaches-it.htmlSuppose there is a Carter-Rodino ticket? Some Republicans might charge that Rodino's chairmanship of the Judiciary Committee in the impeachment of Nixon was tainted by political ambition; but I doubt that many swing voters would take that seriously. All in all, I think that the only likely effect of the choice in 1976 would be that Carter might carry NJ, which he lost rather narrowly in OTL.
https://en.wikipedia.org/wiki/United_States_presidential_election_in_New_Jersey,_1976The real effect comes in 1984: Mondale is a lot less likely to run for president (or to win the nomination if he does run) if he had never been elected VP. The nomination of Gary Hart therefore seems a lot more likely. (Rodino is probably too old to run for POTUS in 1984.)
(One other consequence would presumably be that NJ-10--which as of 1976 was basically Newark plus East Orange--would get an African American Representative twelve years earlier than in OTL. But who? In 1972 East Orange Mayor William S. Hart challenged Rodino and did respectably--he got 36.78% of the vote to Rodino's 57.42%.
https://www.ourcampaigns.com/RaceDetail.html?RaceID=236178 But Hart would be unlikely to run in 1976, having been indicted for bribery, though acquitted in late July.
https://www.nytimes.com/1976/07/23/archives/new-jersey-briefs-hart-is-acquitted-of-bribe-charge-picatinny-funds.html Another possibility is Donald Payne, who challenged Rodino in 1980 and 1986 and ultimately succeeded him in 1988.
https://en.wikipedia.org/wiki/Donald_M._Payne)