The 60-year-old, whose daughter died of cancer in 2011, lost an action at the High Court last year, but is now asking three Court of Appeal judges to rule in her favour.
Her QC, Jenni Richards, told the judges on Wednesday that the woman wants to fulfil her daughter's wishes that her mother should carry a child created from her frozen eggs "and raise that child".
Ms Richards said that if the court did not overturn the High Court's ruling, the "inevitable" consequence would be that the eggs "will simply be allowed to perish".
The woman and her 59-year-old husband are challenging the decision of Mr Justice Ouseley last June to dismiss their case.
During the High Court proceedings, the judge was told that the daughter, who can only be referred to as ''A'' for legal reasons, was desperate to have children and asked her mother to ''carry my babies''.
Her parents, who are referred to as "Mr and Mrs M", launched legal action against an independent regulator's refusal to allow them to take their daughter's eggs to a US fertility treatment clinic to be used with donor sperm.
The Human Fertilisation and Embryology Authority (HFEA) said the eggs could not be released from storage in London because A did not give her full written consent before she died at the age of 28 of bowel cancer.
Mr Justice Ouseley heard that A would have been ''devastated'' if she had known her eggs could not be used.
But he ruled that the HFEA had been entitled to find the daughter had not given ''the required consent''. He declared there had been no breach of the family's human rights.
He said: ''I must dismiss this claim, though I do so conscious of the additional distress which this will bring to the claimants, whose aim has been to honour their daughter's dying wish for something of her to live on after her untimely death.''
Ms Richards argues that there is "clear evidence" of what A wanted to happen to her eggs after she died, and that "all available evidence" showed she wanted her mother "to have her child after death".
She told Sir James Munby, president of the High Court's family division, sitting in London with Lady Justice Arden and Lord Justice Burnett, that the regulator's refusal decision was not based on "any matter relating to the age of Mrs M, or family connection, or any child welfare issues".
She said the case was not about "scientific or ethical principles". The decision was based solely on an "evaluation of the evidence relating to A's wishes".
The QC said: "This court has a duty to decide for itself whether the decision strikes a fair balance between the rights of the individual and the interests of the community as a whole."
The appeal is opposed by the HFEA, which argues that Mr Justice Ouseley "did not err in concluding that the HFEA's decision was lawful".