My husband has a son to a woman who died a few weeks ago he's 2 yrs old and my husband is in the middle of a custody battle with the child's grandparent, there's no doubt he will win we know 100%, the child has already forgotten about her he's so young I am happy to bring him up and treat him like my own what is the ruling on adoption after all he's my husbands child not a stranger and also I wanted to know by what name would he have to call me by?
first, LOVE & take care of that little boy. He will thank you when he grows up.
In Arabic, it's common for Husband's children to call their father's wife (that's you) an aunt. Technically, An aunt in Arabic refers to only someone's mother's sister(s) rather than both father's sisters and mother's sisters (as it is used in English).
He can call you step-mom (in English) which means Father's wife in both Arabic and English. In Arabic, calling a father's wife an aunt is seen as more respectful than just "father's wife (step-mom)". Being called an aunt means being part of the family. On the other hand being called "father's wife" sounds like it is just a fact, nothing more nothing less.
Also, since you are married to his father, he becomes Mahrah. However, unlike your own children, in Islam he is allowed to marry people that your own children are not such as your sister (your children's aunt) because he's not considered as Mahrah to your family (sisters, mother, etc). Also, he does NOT inherit you (get some of your money) when you pass away.
Technically, in your case, you do not need to ADOPT your husband's child. You are married to his father and therefore you can call him "my son" even if his mother was alive; while he is not your biological son, Islam treats him this way in many aspects: your babies with his father are his siblings and he is Mahrah to you and them.
Islam allows adoption (adopting a baby that is not your husband's) with some restrictions such as having him (baby) keep his father's last name. The Prophet (pbuh) ADOPTED Zayd ibn Harithah and called him (Zayd ibn Muhammad) but Allah asked the Prophet NOT to call him ibn Muhammad (son of Muhammad) but let the boy keep his father's last name ibn Harithah (son of Harithah).
Legally, I'm not sure how that works. However, giving that your husband's baby will be keeping his father's last name (your husband) and living with both of you, you being called his mom (in legal terms) does not matter in Islam.
Final note: while keeping the father's last name is a requirement in Islam, it is not always possible to do so. What if the baby's father is NOT known. In that case, I think Islam would say OK to give the baby his adopting father's last name.