Young children from around the world are accessing the internet in ever increasing numbers. The rapid increase in internet activity by children aged 4-5 years in particular is due to the ease access enabled them by touchscreen internet-enabled tablet technologies. With young children now online, often independently of adult supervision, the need for early childhood cyber-safety education is becoming urgent. In this paper, we report the early findings from a project aimed at examining the development of cyber-safety education for young children. We argue that cyber-safety education for young children cannot be effectively developed without first considering young children's thinking about the internet. In this paper, we use Vygotsky's ideas about the development of mature concepts from the merging of everyday and scientific concepts. We identify the potential range of everyday concepts likely to form the basis of young children's thinking about the internet as a platform for cyber-safety education in the early years.