Is it necessary to do two Tawaafs, one for the child and one for the person who is carrying him, or is one Tawaaf sufficient for both?
Question

                I have decided to do Hajj and I have a small child with me. Do I have to do Tawaaf for myself and then do another Tawaaf for him, or will one Tawaaf and one Sa’ee be sufficient?
Praise be to Allah.In the name of Allaah, the most Gracious, 
  the most Merciful  
Answer: the scholars are agreed that the 
  Hajj of a child is valid. Abu Haneefah said: he is not subject to the obligations 
  of expiation. The scholars are agreed that this does not count as the obligatory 
  Hajj, and he must do another Hajj after he reaches adolescence.   
The Hajj of a child may be one of the following 
  three cases:  
1.        
  Where the child is able to walk, in which case he does Tawaaf and Saa’ee 
  on his own behalf. 
2.       
  Where he is not able to walk, but he has reached an age of understanding. 
  In this case both the one who is carrying him and the one who is carried (i.e., 
  the child) should have the intention on his own behalf, then one Tawaaf and 
  one Sa’ee will be sufficient for both of them. 
3.         
  Where the child is very young and has not reached an age of understanding. 
  In this case his guardian or someone else should carry him and make the intention 
  on his behalf. Then one Tawaaf and one Sa’ee will be sufficient for both 
  of them. Their case is similar to that of one who is riding.  
Some of the scholars said that he should 
  do Tawaaf on his own behalf then do another Tawaaf on behalf of the child. 
But the first view is the one which is correct. It was 
  narrated in Saheeh Muslim (1336) via Ibn ‘Uyaynah from Ibraaheem 
  ibn ‘Uqbah from Kareeb the mawla (freed slave) of Ibn ‘Abbaas 
  from Ibn ‘Abbaas that the Prophet  
  (peace and blessings of Allaah be upon him) met a group of people in al-Rawhaa’, 
  and said, “Who are you?” They said, “Muslims.” They 
  said, “Who are you?” He said, “The Messenger of Allaah.” 
  A woman lifted up a small child and said, “Is Hajj counted for this one?” 
  He said, “Yes, and you will have the reward.”  
The Prophet  
  (peace and blessings of Allaah be upon him) did not say, do two Tawaafs, one 
  on his behalf and one on your own behalf. It is not permissible to delay explaining 
  something at the time when it is needed.  
This is the view of Abu Haneefah and was favoured by Ibn 
  al-Mundhir. Abu Muhammad ibn Hazam (may Allaah have mercy on him) said in al-Muhallaa 
  (5/320): “We regard it as mustahabb to take a child for Hajj even if he 
  is very young, or he  is old enough. This will count as Hajj for him and 
  he will have the reward, but it is a voluntary Hajj. The one who takes him for 
  Hajj will also have reward. He (the child) should avoid all the things that 
  the muhrim should avoid, but there is no penalty on him if he does anything 
  that is not allowed. He may be carried for Tawaaf and the Ramiy (throwing stones 
  at the Jamaraat) may be done on his behalf if he is not able to do it. The Tawaaf 
  of the one who takes him for Tawaaf will count as Tawaaf for both of them…”  
  
Because there is no difference between 
  this and the one who is riding. The Tawaaf counts for both the carrier and the 
  one who is carried. And Allaah knows best.  
Shaykh Ibn Baaz (may Allaah have mercy 
  on him) said: If the carrier intends to do tawaaf and sa’ee both on his 
  own behalf and on behalf of the person who is being carried, that will count 
  for both of them, according to the more sound of the two opinions. Because the 
  Prophet  (peace and blessings of Allaah 
  be upon him) did not tell the woman who asked him about the child’s Hajj 
  to do another Tawaaf for him. If that had been obligatory, he  
  (peace and blessings of Allaah be upon him) would have said so.  
See Majmoo’ Fataawa Shaykh ‘Abd 
  al-‘Azeez ibn Baaz, 5/257   
Shaykh Ibn Jibreen was asked about that 
  and he said:  
The ihraam of a child is valid, but his guardian is responsible 
  for him. So he should dress him for ihraam and tie the ihraam garments on him, 
  and make intention for Hajj on his behalf, and recite Talbiyah on his behalf, 
  then he should hold his hand during Tawaaf and Sa’ee. If he is incapable 
  of doing that, as in the case of a very small child or nursing infant, then 
  there is nothing wrong with him carrying him. One Tawaaf is sufficient for both 
  the carrier and the one who is carried, according to the correct view…  
  
See Fataawa Islamiyyah, vol. 2, p. 
  182