What is the ruling on designating a place in the house for worship and prayer?

There is no objection to designating a place in the house for worship and prayer. The Sunnah (definition) indicates the permissibility of this, as reported from Maymunah, the wife of the Prophet (peace and blessings be upon him): "She would be menstruating and not praying, and she would be spread out in front of the mosque of the Messenger of Allah (peace and blessings be upon him), and he would be praying on his small prayer rug (khumrah)." And from Utban ibn Malik, who said to the Prophet (peace and blessings be upon him): "Then pray, O Messenger of Allah, in my house, in a place I can take as a place of prayer." So the Messenger of Allah (peace and blessings be upon him) came to him and said: "Where would you like me to pray?" So he pointed to a place in the house, and the Messenger of Allah (peace and blessings be upon him) prayed there.

What is meant by a "mosque of the house" is a place within it designated for prayer. These places do not take on any of the rulings of dedicated mosques (masajid musabbalah). Rather, these spots should be respected, kept clean, and purified.