What is the ruling on making up eight days from a missed Ramadan, knowing that I am ill and thin and feel tired and nauseous, and my mother advises me to offer expiation for them instead of making them up?

You must make up the eight days before the coming Ramadan (definition) if you are able. If you are unable due to illness, there is no harm in delaying until after the coming Ramadan. But if you delay without a valid excuse, you must make up the fasts, along with feeding a poor person for each day you delayed making up, according to the majority of scholars.