A woman’s prayer in her house is better than her prayer in the mosque. It was narrated from Umm Humayd, the wife of Abu Humayd al-Saa’idi (may Allaah be pleased with her), that she came to the Prophet (peace and blessings of Allaah be upon him) and said: O Messenger of Allaah, I love to pray with you. He said: "I know that you love to pray with me, but praying in your house is better for you than praying in your courtyard, and praying in your courtyard is better for you than praying in the mosque of your people, and praying in the mosque of your people is better for you than praying in my mosque." So she ordered that a prayer-place be built for her in the furthest and darkest part of her house, and she always prayed there until she met Allaah (i.e., until she died). Narrated by Ahmad (26550); classed as saheeh by Ibn Khuzaymah in his Saheeh (3/95), Ibn Hibbaan (5/595) and al-Albaani in Saheeh al-Targheeb wa’l-Tarheeb (1/135).
It was narrated from ‘Abd-Allaah ibn Mas’ood (may Allaah be pleased with him) that the Prophet (peace and blessings of Allaah be upon him) said:
"A woman’s prayer in her house is better than her prayer in her courtyard, and her prayer in her bedroom is better than her prayer in her house." Narrated by Abu Dawood (570) and al-Tirmidhi (1173); classed as saheeh by Shaykh al-Albaani in Saheeh al-Targheeb wa’l-Tarheeb (1/136).
“In her house” refers to the room in which the woman is.
“in her courtyard” refers to the central part of the house to which all the doors open. It is like what people nowadays call the hallway.
“her bedroom” refers to a smaller room inside a larger room, where she keeps her precious belongings.
See: ‘Awn al-Ma’bood.
Based on this, tell your wife that this is better, and that by praying in the house she will attain a reward greater than that for praying in the mosque, praise be to Allaah.
Your praying on behalf of your wife is not valid, and does not count instead of her own prayer. No one can pray on behalf of anyone else, whether they are living or dead, according to the majority of scholars, and some narrated that there was consensus on this point. See: Fath al-Baari (11/584). It says in al-Mawsoo’ah al-Fiqhiyyah (2/334): As for purely physical acts of worship, such as prayer and fasting, it is not permissible to delegate someone else to do them when one is alive. End quote.
And Allaah knows best.