The Insomnia severity index, translated Swedish and Chinese versions, have been utilized in children and adolescents. Some psychometric properties (please see below) were described in these populations. I can not recall paediatric studies with the original (English) version of the measure.
References:
Kanstrup, M, et.al. Insomnia in paediatric chronic pain and its impact on depression and functional disability. Eur J Pain, Vol. 18 (8): 1094-102, 2014.
ISI showed satisfactory psychometric properties in this sample, including internal consistency and concurrent criteria validity.
Chung, Ka-Fai, et.al. Assessing insomnia in adolescents: comparison of Insomnia Severity Index, Athens Insomnia Scale and Sleep Quality Index. Sleep Med, Vol. 12 (5): 463-70, 2011.
The Cronbach's alpha of ISI was 0.83 and the 2-week test-retest reliability was 0.79. The area under curve of ISI was 0.85, the optimal cut-offs for ISI was a total score of nine (sensitivity/specificity: 0.87/0.75).