Arguments with family members, even over seemingly "small" issues, are often accompanied by resentments, history and habitual patterns of communication. Because of this, apologizing to relatives, including your mother, may be much more difficult than apologizing to friends and acquaintances. However, a sincere and intentional apology can be a powerful tool in repairing relationships and changing ineffective family communication styles.
Examine your intention before apologizing. If your apology is not based on a true desire to express your regret and take responsibility for your part in the disagreement, it will not feel meaningful to you, and may cause you to have resentments. It will also feel insincere to your mother.
Verbalize a statement of regret at the start of your apology. In this portion of your amends, you should express your awareness of how you have hurt your mother. As you do this, you also let her know that you regret and are sorry for your impact on her. An example might be, "I am so sorry I hurt your feelings."
Take responsibility for your actions. Do this by clearly and specifically voicing the areas of the argument in which you accept accountability. An example might be, "I am sorry I said you were never there for me. I was trying to hurt you and I know what I said is not true."
Offer a remedy to repair the damage you have caused. The remedy can be as simple as a sincere promise to not repeat the behavior. You should choose a remedy that will be meaningful to your mother and one you are willing to complete and are capable of accomplishing.