Your aunt is very likely eligible for a British passport. She would be a
British Overseas Territory Citizen (not to be confused with British Overseas Citizen, which is a different category of British nationality with far fewer rights) by descent from her father, based on
Section 23 of the British Nationality Act 1981. All BOTCs were given British citizenship in 2002, so she should be able to apply for a British citizen passport directly.
To apply for this passport, she will need her father's St Helena birth certificate, her parents marriage certificate and her own birth certificate.
Note that many UK social benefits are based on residing in the UK and having paid taxes here. So, for a start, your aunt will not be eligible for the State Pension, which requires a minimum of 10 years of National Insurance contributions (35 years for a full State Pension). Even the NHS requires the British citizen to be "ordinarily resident" in the UK. So she may not be eligible for NHS coverage from day one.
And note that she will herself be a British citizen by descent. So her children will not benefit from her UK citizen status. And she would need to meet all the financial requirements (minimum annual income of £29,000, etc) if she plans to sponsor a spouse to the UK.
I am not a lawyer or immigration advisor. My statements/comments do not constitute legal advice. E&OE. Please do not PM me for advice.