Introducing kindergarten children to the engineering design process (EDP) is an important objective of early STEM education. Studies indicate that children often miss the crucial steps of testing and optimising during the EDP and do not persist in making solutions better. The present study explores how children's goal awareness, self-evaluation ability, domain-specific content knowledge, spatial skills and intelligence relate to their persistence, testing and optimising behaviour and to solution quality. In a standardized procedure, 41 children (4 to 7 years) in Germany worked on an engineering task in the domain of gears. The engineering process was videotaped, children's testing and optimising, solution quality, goal awareness, self-evaluation, and task persistence were rated with a coding scheme and with interviewer questions. Domain-specific content knowledge, mental rotation ability and figural reasoning were measured with standardized tests. Correlational analyses indicated that goal awareness was positively related to solution quality. However, most children required support by the interviewer to retrieve the goal specifications. Moreover, children's self-evaluation was negatively related to task persistence. Most children were satisfied with their first solution, even when it did not meet the requirements. Our findings emphasize the important role of teachers in helping children to tackle challenges with the EDP.