Yes, a baby's হজ্জ is valid and the parent earns reward — use a carrier instead of a stroller, plan for cooler hours, and bring all supplies.
Yes, performing হজ্জ with a baby is permissible. A woman brought her child to নবী করীম (সাল্লাল্লাহু আলাইহি ওয়া সাল্লাম) and asked: 'Is there হজ্জ for this one?' He replied: 'Yes, and you will have the reward' (Muslim). The হজ্জ performed on behalf of a child is valid and earns reward for the parent, though the child will still need to perform their own ওয়াজিব হজ্জ after reaching puberty.
Practical considerations are significant: the extreme heat and massive crowds during হজ্জ present real challenges with an infant. Use a front-facing baby carrier rather than a stroller for তাওয়াফ and sa'i, as strollers are nearly impossible to navigate in peak crowds. Bring ample supplies — diapers, formula/food, medications, sun protection, and a lightweight blanket. Plan your rituals during the early morning or late evening when temperatures are lower and crowds are thinner. Keep the baby hydrated frequently. Diaper changes and breastfeeding breaks may require stepping away from the ritual temporarily — this is completely permissible. Ensure your হজ্জ group is aware you have an infant so they can provide appropriate support.
Source: Sahih Muslim, Kitab al-হজ্জ; Ibn Qudamah, al-Mughni