Is it permissible for a plot of land to be registered in the wife's name for her to build her house on it, given that she is employed, while the husband wishes to buy the remaining part of the land and register the house in his name, despite him having borrowed from his father to complete the purchase?

It is permissible, according to Islamic law, for a woman to register her properties in her own name. This is preferable to avoid disputes and preserve harmony, as a woman has her own independent financial autonomy and the right to dispose of her wealth, just like a man. It is not permissible for her husband or anyone else to take her money without her consent and willing acceptance, based on the Almighty's saying: ﴿But if they give up willingly to you anything of it, then consume it with pleasure and ease.﴾ [An-Nisa: 4], and His saying: ﴿And it is not lawful for you to take anything of what you have given them,﴾ [Al-Baqarah: 229], and the Prophet's (peace be upon him) saying: "The property of a Muslim is not lawful except with his willing consent."

As for a woman registering her properties in her husband's name, if it is by way of a gift and she is of sound mind, then there is no objection. However, if it is merely a nominal registration while she retains real ownership, it should not be done except out of necessity, as it may lead to disputes and problems.