This is from the Social Security Administration:
Changing your name on your Social Security card
If you change your name, be sure to tell both Social Security and your employer. This will assure that your earnings will be properly reported by your employer and recorded in our records. You can get a new card from Social Security with your new name.
How divorce affects your future retirement benefits
If you are divorced after at least 10 years of marriage, you can collect retirement benefits on your former spouse’s Social Security record if you are at least age 62 and if your former spouse is entitled to or receiving benefits. If you remarry, you generally cannot collect benefits on your former spouse’s record unless your later marriage ends (whether by death, divorce, or annulment).
How divorce affects survivors benefits
If your divorced spouse dies, you can receive benefits as a widow/widower if the marriage lasted 10 years or more. Benefits paid to a surviving divorced spouse who is 60 or older will not affect the benefit rates for other survivors receiving benefits.
How remarriage affects survivors benefits
In general, you cannot receive survivors benefits if you remarry before the age of 60 unless the latter marriage ends, whether by death, divorce, or annulment. If you remarry after age 60 (50 if disabled), you can still collect benefits on your former spouse’s record. When you reach age 62 or older, you may get retirement benefits on the record of your new spouse if they are higher. Your remarriage would have no effect on the benefits being paid to your children.
How retirement affects survivors benefits
If you are collecting survivors benefits, you can switch to your own retirement benefits (assuming you are eligible and your retirement rate is higher than the widow/widower’s rate) as early as age 62. In many cases, you can begin receiving retirement benefits either on your own or your spouse’s record at age 62 and then switch to the other benefit when you reach full retirement age, if that amount is higher.