Is it permissible for my mother to perform tayammum for Fajr prayer because she cannot tolerate using water immediately upon waking, even though she does not have a specific illness, and what are the easiest correct ways to perform tayammum, and is it permissible to wipe over socks during wudu inside the house for ease?

Illness or lack of water permits tayammum (definition), as Allah Almighty says: "And if you are ill or on a journey or one of you comes from the place of relieving himself or you have contacted women and find no water, then seek clean earth and wipe over your faces and hands [with it]."

If your mother does not have an illness that prevents her from using water, and it is only the Fajr prayer that is difficult for her, then tayammum is not permissible for her. Rather, she must perform ablution, which carries great reward.

To make it easier for her, she can perform ablution inside the house, warm the water, dry her limbs after ablution, stay indoors for a short period, and use ointments on the ablution limbs. It is also permissible for her to wear socks and wipe over them after performing ablution.