Weah, 51, won the first round, but did not secure the required 50% of the vote for an outright victory.
The second round of voting to succeed Ellen Johnson Sirleaf was delayed because of legal challenges.
Liberia, which was founded by freed US slaves in the 19th Century, has not had a smooth transfer of power in 73 years, says a BBC report.
Polls across Liberia will open at 08:00 GMT and close at 18:00 GMT.
More than two million people are eligible to cast their ballots.
Boakai, 73, has been the Liberian vice-president for 12 years - but does not seem to enjoy the support of his boss.
The candidate has described himself as being like a limousine idly parked in a garage all these years, who is now ready to hit the road.
Meanwhile, Weah, the 1995 world footballer of the year, is hoping for third time lucky.
He defeated Johnson-Sirleaf in the first round in 2005, but lost to her in the subsequent run-off. In the following election, as a presidential running mate, his coalition boycotted the run-off, citing irregularities.
A representative for the opposition Liberty Party, Charles Brumskine, who came third in October's first round, challenged the result saying it had been marred by "massive fraud and irregularities".
But earlier this month, the Supreme Court ruled that evidence of fraud was insufficient to merit a re-run.
The verdict meant that the run-off - which had initially been set for 7 November - could go ahead.
This will be the first time for many generations in Liberia to witness a transfer of power from one elected leader to another.
Outgoing President Sirleaf has praised a violence-free electoral process, saying that "the ballot box has replaced bullets and electoral disputes are settled through the courts".
Sirleaf took office in 2006, after her predecessor, Charles Taylor, was forced out by rebels in 2003, ending a long civil war.
Taylor is currently serving a 50-year prison sentence in the UK for war crimes related to the conflict in neighbouring Sierra Leone.
The results of the run-off are due to be announced within four days.