Ms Hewitt also said the government will consider removing the health secretary from the process that sees councillors referring service changes to the Independent Reconfiguration Panel.
And she suggested 'Bank of England-style independence' for the Advisory Committee on Resource Allocation, which determines the funding formula for PCTs.
Speaking at the London School of Economics, Ms Hewitt said: 'In future, we may see some local PCTs and their residents decide some PCT board members should be directly elected. But developments must not be imposed from above; they should grow organically as PCTs and local authority partners become more confident in their own roles and co-operation between them.'
Referring to the need to 'balance competition', Ms Hewitt said the 'next step forward should be to involve staff and patients in developing a new statement of NHS values that could then be enshrined in a new NHS constitution'.
Her comments on the panel come after the Institute for Public Policy Research criticised the current system. It said letting the minister decide whether to refer matters from council overview and scrutiny committees to the panel undermined public confidence by appearing 'politically motivated'.
Councillors should instead be able to send a contested reconfiguration to the panel - although Ms Hewitt said not all matters would necessarily undergo full investigation by the panel.
She said in last Thursday's speech: 'We should now consider separating [the panel] from government, so that overview and scrutiny committees could refer a proposal directly to the panel rather than to the health secretary.
'The panel would have to weed out the weak referrals, which the [Department of Health] does for them at the moment, but I'm sure they could cope.'
Ms Hewitt continued: 'We should also consider Bank of England-style independence for the Advisory Committee on Resource Allocation.'
The health secretary should still 'set the goal' for the body but leave it to carry out 'the exhaustive statistical analysis that no minister could, or should, try to get involved in'.
Ms Hewitt poured cold water on calls for an independent board for the NHS. She said: 'Putting a doctor, a manager or a patients' champion at the head of an independent NHS board might be more popular than having a politician in charge, but as soon as the board started making difficult decisions the attraction would wear off and the public would rightly ask what the difference was.'
But steps 'towards creating an independent, self-improving NHS' had been taken with reforms such as foundation trusts and the National Institute for Health and Clinical Excellence, she said.
Although Labour had 'saved' the NHS, the 'public as a whole is not yet persuaded'. She said: 'There is no doubt that recent years have left bruises, particularly on staff morale, but also on public confidence.'
Ms Hewitt added: 'I know there have been tough decisions around cuts to training budgets and deferred developments, but underneath all that the NHS treated more patients, more quickly, with better outcomes and higher patient satisfaction than in previous years.'