What is the ruling on naming someone "Amirat Ar-Rahman" (Princess of the Most Merciful) or "Amirat Saif Ar-Rahman" (Princess of the Sword of the Most Merciful)?

There is no objection to naming a newborn girl "Amirat al-Rahman" (Princess of the Merciful) because the attribution to Allah here is an attribution of creation and ownership to Him, Glorified and Exalted, and this encompasses all creation. Ibn Taymiyyah mentioned that when entities are attributed to Allah, it is either a general, shared attribution (as something created and owned by Him), or a specific attribution unique to it, such as "Baytullah" (House of Allah) and "Naqatullah" (She-camel of Allah). However, it is preferable to name the newborn "Amat al-Rahman" (Female Servant of the Merciful) because it carries the same meaning as "Abd al-Rahman" (Servant of the Merciful), which is one of the names most beloved to Allah.