But that also saddles him with the legacy of his brother's hawkish policies in the Middle East, particularly the 2003 Iraq war which most of the American public still believes was an historic mistake.
To that end, the Florida governor has spent this past month in a delicate balancing act of attempting to define his foreign policy as being distinct from his brothers, without being disloyal.
It is a challenge he is struggling to meet, if the polls are anything to go by: one survey of Florida voters last week showed him falling behind newly-entered Republican candidate Donald Trump, with Trump on 21 per cent of the vote and Bush slipping from 20 per cent in June to 17 per cent.
"He has to reach voters in a way where they are not looking at his candidacy through the lens of his brother's presidency," said Kevin Madden, a Republican strategist who has worked on past campaigns. "That is his challenge."
With the polls putting national security as a high priority for voters, Mr Bush has been trying to set out his strategy for defeating Islamic State militants in Iraq and Syria.
In a recent speech, he adopted an approach that seemed calculated to be more hawkish than that of President Obama, but not so much as to make him the third Bush president to deploy thousands of American soldiers to Iraq.
He said he would embed US Special Forces with Iraqi troops units and increase military aid to the Sunni and Kurdish militias fighting Isil. In neighboring Syria, he has promised to build up a domestic army of non-extremist groups, and establish "safe zones" and "no fly zones" to defeat both Isil and President Bashar al-Assad.
But the speech also worked hard to avoid laying blame at his brother's door. Instead he pinned the rise of Isil on the "void" that was left after the pullout of American troops in 2011, when Hillary Clinton, his Democratic rival, was secretary of state
Mr Bush's supporters argue that of two frontrunners, it is Mrs Clinton who should be struggling most with her war record
"I think Jeb Bush is in a better position than Mrs Clinton," said Peter Feaver, who served on the National Security Council under President George W Bush."He has to answer the awkward question of what happened on his brother's watch, and Hillary has to answer the awkward question of what happened on her watch."
On issues other than Iraq, George W Bush still has favourable ratings with much of the American public, and is still potentially a powerful weapon in the aspirational president's arsenal.
But the former Florida governor's refusal to outrightly condemn some of the darker episodes of his brother's tenure - such as the use of waterboarding by the CIA - has been touted by his detractors as "proof" that to elect him would be to re-instill his brother's approach to foreign policy.
The strain has started to show. When The Sunday Telegraph pressed Mr Bush during a recent encounter on whether he would deploy American troops to the Middle East, he was visibly annoyed.
Another problem is that his campaign is staffed with people who also worked for his brother and for his father. "The problem is that the people writing Jeb's speeches are the same people who were involved in the Iraq war," said one Republican strategist.
"It's not that he can't distance himself from his brother. It's that the speechwriters can't distance themselves from the s**t they were writing five years ago."
