"Eating Organic Lowers Pesticide Levels in Children" went the headline reporting on the small study by Asa Bradman and colleagues  (open-access available here). Detailing what happened to urinary pesticides levels following trials of combinations of conventionally grown food consumption vs. organic food consumption over 16 days, researchers reported some potentially interesting findings.
Measuring 23 metabolites "reflecting potential exposure to organophosphorous (OP), pyrethroid, and other pesticides used in homes and agriculture" via everyone's favourite analytical method (LC-MS) and specifically "tandem mass spectrometry", Bradman et al observed that: "An organic diet was significantly associated with reduced urinary concentrations of nonspecific dimethyl OP insecticide metabolites and the herbicide 2,4-D in children." If I had a beef with any part of the Bradman study outside of the fairly small participant group, it would be that the reliance of urinary excretion of pesticide residues might not necessarily show the whole story, as per what data one might get from the use of other biofluids such as blood samples too or even analysis of fat biposies (recognising how invasive these can be).
These are interesting findings added to other similar research on this topic  looking at adults. Not only do they point to the idea that there is persistent low level exposure to pesticide residues in food but also that changes in food consumption patterns may affect such exposure events. That's not to say that food is the only way that pesticide exposure might occur, as per the findings of differences among children living in urban vs. agricultural communities (where those living in more agricultural areas generally had higher levels of some of the more frequently detected pesticide metabolites). But dietary change encompassing an organic diet did seem to lead to reductions in certain pesticide metabolite excretions irrespective of geography.
Accepting the often valuable reasons why pesticides are used in the first place, I don't think many people would argue with the idea that pesticide exposure should be limited, particularly in respect of children and their developing bodies and minds . Allied to the idea that organic food might also confer other benefits in terms of nutritional quality (see here) and lower levels of fairly toxic metals such as cadmium , there seems to be common sense in rethinking some aspects of agriculture for certain groups. That the genetics of pesticide metabolism may also play a role  as per discussions about PON1 (paraoxonase/arylesterase 1), is also an important point and how childhood relates to PON1 activity among other factors . Indeed, to answer the question posed in another paper  yes, there may indeed be some benefits from an organic diet for children...
Music: Adele - Rolling in the Deep.
 Bradman A. et al. Effect of Organic Diet Intervention on Pesticide Exposures in Young Children Living in Low-Income Urban and Agricultural Communities. Environ Health Perspect. 2015 Oct;123(10):1086-93.
 Oates L. et al. Reduction in urinary organophosphate pesticide metabolites in adults after a week-long organic diet. Environ Res. 2014 Jul;132:105-11.
 Muñoz-Quezada MT. et al. Neurodevelopmental effects in children associated with exposure to organophosphate pesticides: a systematic review. Neurotoxicology. 2013 Dec;39:158-68.
 Barański M. et al. Higher antioxidant and lower cadmium concentrations and lower incidence of pesticide residues in organically grown crops: a systematic literature review and meta-analyses. Br J Nutr. 2014 Sep 14;112(5):794-811.
 Gonzalez V. et al. Cholinesterase and paraoxonase (PON1) enzyme activities in Mexican-American mothers and children from an agricultural community. J Expo Sci Environ Epidemiol. 2012 Nov;22(6):641-8.
 Vania A. et al. Is organic diet really necessary for children? Italian Journal of Pediatrics 2015, 41(Suppl 2):A75
Bradman, A., Quirós-Alcalá, L., Castorina, R., Schall, R., Camacho, J., Holland, N., Barr, D., & Eskenazi, B. (2015). Effect of Organic Diet Intervention on Pesticide Exposures in Young Children Living in Low-Income Urban and Agricultural Communities Environmental Health Perspectives, 123 (10) DOI: 10.1289/ehp.1408660