Praise be to Allah.
Firstly: If these birds are a kind whose flesh it is permissible to eat according to sharee’ah, such as sparrows, chickens and ducks and so on, then their faeces is taahir (pure). The same ruling applies to any animal whose flesh may be eaten, such as sheep, cows, horses and so on. There is a great of evidence that the urine and dung of animals whose flesh may be eaten is taahir, such as the following:
1 – The basic principle concerning all things is that they are taahir, and there is no saheeh shar’i evidence to suggest that these things are naajis (impure).
2 – It is proven in al-Saheehayn that the Prophet (peace and blessings of Allaah be upon him) told a group of people who came to Madeenah and fell sick to drink the urine and milk of camels. If the urine of camels was impure he would not have told them to drink it, because it is not permissible to use haraam things as medicine.
3 – It is proven in Saheeh Muslim that the Prophet (peace and blessings of Allaah be upon him) was asked about praying in sheep pens and he said: “Pray in them for they are a blessing.” And he did not tell the one who prayed there to avoid the urine and dung, even though he would usually find some of that.
There is a great deal of other evidence, which was discussed at length by Shaykh al-Islam Ibn Taymiyah (may Allaah have mercy on him). See Majmoo’ al-Fataawa (21/542-586).
Ibn Qudaamah (may Allaah have mercy on him) said in al-Mughni (2/492):
The urine and faeces of animals whose flesh may be eaten is taahir. … Maalik said: The scholars do not regard the urine of animals whose meat is eaten and whose milk is drunk as najis (impure). … Ibn al-Mundhir said: Everyone from whom we acquired knowledge among the scholars is unanimously agreed that it is permissible to pray in sheep pens, apart from al-Shaafa’i who stipulated that it should be free of dung and urine. End quote.
It says in Fataawa al-Lajnah al-Daa’imah (6/414):
The urine of animals whose meat may be eaten is taahir, and if it is used on the body for some purpose, there is nothing wrong with praying in that state. End quote.
But if these birds are of a type whose flesh may not be eaten, such as birds that have talons, like falcons, then their faeces is najis and there is no difference of opinion among the scholars concerning that. See al-Mughni (2/490).
As it is proven that the faeces of birds whose meat may be eaten is taahir, then it does not have to be washed off if it gets onto one’s clothes or body, or on the carpet, and there is nothing wrong with praying in these clothes or on this carpet.
But you should advise your mother to allocate a special place for these birds so that they will not bother other members of the household, because even if this faeces is taahir, it is still off-putting to people.
And Allaah knows best.