All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger.
In principle, it is permissible to name a new-born child “Muhammad Hussain” after his grandfather, but you should know that the scholars stated that naming the child is the right of the father and not the right of the mother, and Ibn Al-Qayyim stated in his book entitled “Tuhfatul Mawdood Fi Ahkaam Al-Mawlood” that the jurists agreed on this.
If we presume that your mother-in-law objects regarding naming him with this name, then you should obey her in this as this is being kind to her. If it is possible for all of you to agree on this name or any other name, then this is better.
Moreover, it is permissible to name the child “Muhammad Ali Hamzah”, because it is permissible to name a child with a compound name and we are not aware of any evidence which prohibits this.
For more benefit, please refer to Fataawa 87262 and 82724.
Allaah Knows best.