This excerpt is taken from Sheikh Farooq Rafi Sahib's book Qurbani, Aqiqah, and Ashra Zil-Hijjah.
The one performing Hajj can also offer a sacrifice on behalf of his family members participating in Hajj. The evidence for this is the following hadith.
① Sayyida Aisha Siddiqah (may Allah be pleased with her) narrates:
خرجنا لا نرى إلا الحج فلما كنا بسرف حضت، فدخل على رسول الله صلى الله عليه وسلم وأنا أبكي، فقال : ما لك، أنفست، قلت : نعم، قال : إن هذا أمر كتبه الله على بنات آدم، فاقضي ما يقضي الحاج غير أن لا تطوفي بالبيت قالت : وضحى رسول الله صلى الله عليه وسلم عن نسائه البقر
"We set out with the intention of pure Hajj, and when we reached the place of Sarf, I became menstruating. Then the Messenger of Allah (peace be upon him) came to me while I was crying. He asked: 'What is it? Have you become menstruant?' I replied: 'Yes!' He said: 'Allah has prescribed this matter upon the children of Adam, so do what the one performing Hajj does, except do not perform Tawaf of the House (Kaaba).' Sayyida Aisha (may Allah be pleased with her) narrates that the Messenger of Allah (peace be upon him) slaughtered a cow on behalf of his wives (on the day of Eid al-Adha)."
Reference: Sahih Bukhari, Book of Menstruation, Chapter: The command to the menstruating woman when she menstruates: 294
② Sayyida Aisha (may Allah be pleased with her) narrates:
فلما كان يوم النحر أتيت بلحم بقر كثير فطرح فى بيتي، فقلت : ما هذا ؟ قالوا : ذبح رسول الله صلى الله عليه وسلم عن نسائه البقر
"(During the Farewell Pilgrimage) on the Day of Sacrifice, a large amount of cow meat was brought to me and kept in my house. I asked: 'What is this?' They said: 'The Messenger of Allah (peace be upon him) has slaughtered a cow on behalf of his wives.'"
Reference: Hasan: Musnad Ahmad: 6 / 272
Benefits:
① Hafiz Ibn Hajar, may Allah have mercy on him, states: "It appears evident that the slaughtering of the cow was for sacrifice."
Reference: Fath al-Bari: 10 / 8
② It is Sunnah to offer a separate sacrifice on behalf of family members participating in Hajj.
