But first she needs to win Labour's nomination, which will be decided next summer. A poll last month placed her a close second behind the frontrunner, former Olympics minister Tessa Jowell.
Both women, however, face a dilemma: each are former ministers, closely associated with New Labour, and each can lay claim to an appeal beyond traditional Labour voters.
"I think we are very different people," suggested Ms Hodge citing the fact that she grew up in Orpington, in Kent, and went on to represent Barking.
"I've got this outer London feel, which I think is very important. I've always been much more of a rebel and less of a loyalist than she has. I'm more outspoken, which for the mayor, would be a strength."
Indeed, she cannot quite leave the topic alone. Later, when we are discussing her popularity as PAC chair - "I've never been so popular. It's ironic, isn't it? Taken me 40 years to get there." - she suddenly pops up with: 'I'd like to think that one of my strengths as mayor is that I am slightly detached from the politics."
In 2012, Ken Livingstone's fractious relationship with the capital's Jewish community almost certainly hurt him in his race against Boris Johnson. Ms Hodge did not want Mr Livingstone as the candidate, backing Oona King's bid for the Labour nomination.
There is little love lost between Ms Hodge and Mr Livingstone. She recalls Tony Blair asking her to draw up a paper on an elected mayor for London shortly after Labour came to power in 1997. She went to discuss it with Livingstone. "I thought he would hate it … and he jumped at it and I thought: 'Oh, sugar, he's clearly going to stand'."
At the root of Livingstone's problems with the Jewish community, she believes, rests the fact that he "could never distinguish between being critical of the Israeli government… and being critical of Jews in Britain". But she also hints at something darker.
"He was pretty horrible about me", she said accusing him of making "presumptions about my attitudes to the Israeli state tied up with the fact that I was of Jewish heritage".
Could she heal that breach with London's Jews in 2016? "I hope so, I hope so," she responded.
She admits, though, that some of the Jewish community are "a little wary of me and my commitment". She attributes some of this to the fact that she is not religious and both her marriages were to non-Jews.
It also stems, she argues, from her stance on Israel. She said: "I think you can support the Israeli state and you can support their desire for security. But you can equally say that their reaction has been disproportionate."
She says her views on Israel attract hostility from Jewish audiences and appeals for greater tolerance within the community for views such as hers. "It isn't an attack on Israel, it isn't an attack on the Jewish community if some of us just feel that the direction of travel of the current administration in Israel isn't supporting a long-term peaceful settlement," she said.
She is sad that Maureen Lipman, the actor who has abandoned her support for Labour over Ed Miliband's position on Gaza and Palestinian statehood, has decided to make the issue the litmus test for her vote. Ms Hodge's abstention on the vote on Palestinian statehood was not deliberate. She had a speaking engagement and had she been in parliament, she would have voted for it.
But doesn't Lipman's stance point to a wider alienation of much of the community from Miliband? "For heaven's sake, it's great to have a Jewish leader, isn't it?"
Ms Hodge, who spent time on a kibbutz in the 1960s, insists that her views on Israel are not "a denial of my race and identity, and neither are they an attack on my people". She is, she repeats several times, "Jewish to the core". The way she leads her life is "completely informed by Jewish values around family".
It also has to do with her family's experience of the Holocaust. She was born Margaret Oppenheimer, the daughter of a German-Jewish businessman who left Germany in the 1930s. Her aunt's husband died in Auschwitz, however, and the discovery of a letter from her grandmother, who was shot outside a concentration camp, has been much on her mind recently.
"She'd refused to leave because she was in Vienna, and she thought nobody would attack her because she was too old. She wrote this really, really poignant letter when she knew she was going to be rounded up, and I think clearly knew she was going to die. Full of "don't forget me, don't forget me".' 'When I saw that,' says Hodge, her eyes filling with tears, 'that intensifies my identity with a downtrodden race'.
