Traditionally, courts ruled that a father had an automatic right to have his child keep his last name if he continued to actively perform his parental role. Although there is still some bias in this direction, it is no longer strictly true.
Now a child's name may be changed by court petition when it is clearly in the best interest of the child to do so. When deciding whether to grant a name change, courts consider many factors, such as the length of time the father's name has been used, the strength of the mother-child relationship and the need of the child to identify with a new family unit (if the change involves remarriage). The courts must balance these factors against the strength and importance of the father-child relationship. What this all boils down to is that it's up to a judge to decide which name is in the child's best interest.
Keep in mind that, even if you do change your children's last name, you won't be changing the legally recognized identity of their father. Nor will a name change affect the rights or duties of either parent regarding visitation, child support, or rights of inheritance. Changes such as these occur only if the parental roles are altered by court order -- for example, a new custody decree or a legal stepparent adoption. For more on abandonment of parental rights, see Nolo's article on Stepparent Adoptions.
If your new spouse adopts your child through a stepparent adoption, the child's name can be changed as part of that procedure. If you'd like to get some professional help, see Nolo's Lawyer Directory, where you can search the Family Law attorney listings.