Kennedy originally opened its doors in January of 1961 as Southeast Elementary School. By April of the same year, the Lawrence Board of Education unanimously voted to rename the "new" school, Kennedy Elementary School, in honor of Miss Opal Jayne Kennedy. Miss Kennedy retired as principal of Hillcrest Elementary School in 1961 after 50 years of teaching the children of Lawrence and Douglas County.
A two-year construction project, including the addition of 10 classrooms, concluded in the 2015-2016 school year. Community investment in a school board election funded these facility improvements.
The Lawrence Board of Education voted on April 12, 2021, to repurpose Kennedy Elementary School as an early childhood community center to bring together the district's early childhood programs with community services for young children and their families at one location. Boundary changes transitioned elementary students to Cordley, Prairie Park, and New York schools for the 2021-2022 school year.
Kennedy has housed the early childhood educational readiness program since 2009. The program serves families meeting eligibility requirements, including families of children receiving special education and/or English as a Second Language services. The district also offers services for children from birth to age five through Parents as Teachers and tiny-k Early Intervention.
Head Start also has offered a community preschool classroom at Kennedy.