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.
In most states, you can request that the judge handling your divorce make a formal order restoring your former or birth name. If your divorce decree contains such an order, that's all the paperwork you'll need. You'll want to get certified copies of the order as proof of the name change -- check with the court clerk for details. Once you have this official documentation, you can use it to have your name changed on your identification and personal records.
If your divorce decree doesn't contain an order restoring your former name, check to see if it can be modified to include language restoring your name. In some states, this is possible even after the divorce is final. For example, in California you would use a form entitled Ex Parte Application for Restoration of Former Name After Entry of Judgment of Order (FL-395).
Even if your divorce papers don't show your name change, you may still be able to resume your former name without much fuss, especially if you still have some proof of that name, such as a birth certificate or old passport. In most states, you can simply begin using your former name consistently, and request that it be changed on all your personal records (see Changing Identification and Records).
If you're returning to a name you had before marriage, you're far less likely to be hassled about the change than if you adopt a completely new name, but you may still face some bureaucratic barriers in returning to a previous name. This is especially likely if you are a recent immigrant or do not have reliable documentation of your former name.