Social Security Disability Waiting Period Elimination Act of 1987 - Amends title II (Old Age, Survivors and Disability Insurance) of the Social Security Act to eliminate the five-month waiting period which is a prerequisite of eligibility for widow's or widower's insurance benefits on the basis of a disability and for disability insurance benefits. Eliminates the 24-month waiting period required before individuals may become eligible for hospital insurance benefits under part A (Hospital Insurance) of title XVIII (Medicare) of the Act on the basis of their entitlement to disability benefits under title II of the Act.