The poll was conducted by the Ramallah-based Palestinian Centre for Policy and Survey Research (PCPSR), which interviewed 2,150 adults in the West Bank and the Gaza Strip.
Thirty percent of respondents named Barghouthi, a member of Abbas' Fatah faction whom an Israeli court handed five life sentenced for murder in 2004, as their favourite to replace the president.
Israel accuses Barghouthi, 59, of masterminding attacks by Fatah's armed wing, the Al-Aqsa Martyrs Brigades. Seen as a pragmatist, he enjoys strong grassroots support and has good relations with all factions, including with terror group Hamas.
The next most popular candidate was Ismail Haniyeh, head of Hamas, with 23 percent support.
Questions about Abbas's prospects were raised by his eight-day hospitalisation in May for what officials said was a lung infection.
Wednesday's poll found that 61 percent of the public want Abbas to resign and 33 percent want him to stay in office.
Abbas became president in 2005 after the death of Palestinian leader Yasser Arafat, and pursued US-led peace talks with Israel. But many Palestinians lost faith in him as his efforts yielded no gains and talks stalled in 2014.
Abbas' democratic mandate expired nine years ago, with no new elections set due to a Fatah-Hamas power struggle that has been exacerbated by Israeli controls on the West Bank and the Gaza Strip.
Trailing Barghouthi and Haniyeh as mooted Abbas successors were Prime Minister Rami Hamdallah and former Gaza strongman Mohammad Dahlan with 6 percent support each, West Bank activist Mustafa Barghouthi and former Hamas chief Khaled Meshaal (three percent each), and ex-finance minister Salam Fayyad (two percent).
The PCPSR poll found that 48 percent of Palestinians believe that, post-Abbas, rival factions would be able to agree on a new election; 41 percent said they did not believe this.