There is no minimum length of marriage to be eligible for survivor's benefits.
An excerpt from SS's website:
•Your widow or widower may be able to receive full benefits at full retirement age. The full retirement age for survivors is age 66 for people born in 1945-1956 and will gradually increase to age 67 for people born in 1962 or later. Reduced widow or widower benefits can be received as early as age 60. If your surviving spouse is disabled, benefits can begin as early as age 50. For more information on widows, widowers and other survivors, visit www.socialsecurity.gov/ww&os2.htm.
•Your widow or widower can receive benefits at any age if she or he takes care of your child who is receiving Social Security benefits and younger than age 16 or disabled.
To be eligible for widow(er)'s benefits you generally have to have been married to the deceased worker for at least nine months just before the worker died. See § 401(F):
There are exceptions to this rule: http://www.ssa.gov/OP_Home/handbook/handbook.04/handbook-0404.html
Note, however, the exceptions to the nine-month duration of marriage requirement in A., B., and C. above do not apply if, at the time of the marriage, the insured worker could not reasonably have been expected to live for nine months.
This is a Frequently Asked Question for Social Security. As such, the Social Security Administration has stated the general rule and all the exceptions in one link listed below. According to the SSA, "Generally, a person can qualify for widow's or widower's benefits if he or she was married to the deceased worker for at least nine months just before the worker died. However, you do not need to be married to the worker for any specific length of time if: ..." [see below].