Firstly, some groups of people are not obliged to live in the initial reception centre (Erstaufnahmeeinrichtung) after they make an asylum claim. If your mother has a residence permit that is in total valid for at least 6 months, she has to apply for asylum with the central office of BAMF and does not have to live in a reception centre.
If this is not the case, she will have to live in a reception centre. However, due to her age she would be in a special category of people called ‘vulnerable asylum seekers’. The employees at reception centres are obliged to assess people staying there to see if they are vulnerable. A person your mother’s age should then have their stay shortened to as brief as possible in the reception centre. How long this takes usually depends on how much capacity there is in other more suitable accommodation.
If your mother applies for asylum, then I recommend you communicate to the authority in charge of your local reception centre (this is generally the Sozialamt) that you have suitable accommodation available for her with you and are willing to have her stay with you. It is, however, ultimately at the discretion of the authorities whether to reduce her stay there.
If you are unable to come to an arrangement with the Sozialamt, your mother can make a written application to have her obligation to live in a reception facility removed.
The law which obliges the authorities to take into account the needs of vulnerable people in the asylum process is the EU’s Reception Directive, Article 21 includes elderly people as vulnerable: