Is the son of the husband's wife—who is not his son—entitled to a share of the bequest or inheritance?

The former son of your wife does not inherit from your husband who raised him, because he is not his son. The causes of inheritance are: marriage, lineage, and wala' (filiation through allegiance). It is permissible for your husband to bequeath to him up to one third, and anything exceeding one third can be executed with the consent of the heirs.