In LEO satellite networks with inter-satellite links, the highly dynamic topology and the limited on-board resources pose special challenges to routing protocol design. In this paper, a light weight on-board distributed routing protocol is proposed to cope with these challenges. For ODRP, the single layer LEO satellite constellation is considered as double-layer constellation. A satellite at special geographical position is selected as the plane speaker according to the dynamic characteristics of inter-satellite links and the distribution of traffic load carried by the network, consequently the idea of distributed hierarchical routing is realized. Experimental results show that ODRP has the adaptive abilities to deal with the dynamic topology of LEO satellite networks and guarantees the path's optimality, and especially can decrease the packet loss probability efficiently in case of high traffic load. Furthermore, results from the implementation complexity analysis demonstrate that the proposed protocol has lower onboard computational, storage and signaling requirements than other on-board routing schemes.