IPv6 Low-power Personal Area Networks (6LoWPANs) have recently found renewed interest because of the emergence of Internet of Things (IoT).However, mobility support in 6LoWPANs is still in its infancy for large-scale IP-based sensor technology in future IoT. The hospital wireless network is one important 6LoWPAN application of the IoT, where it keeps continuous monitoring of patients' vital signs while the patients are on the move. Proper mobility management is needed to maintain connectivity between patient nodes and the hospital network to monitor their exact locations. It should also support fault tolerance and optimize energy consumption of the devices. In this paper, we survey IPv6 mobility protocols and propose some solutions which make it more suitable to a hospital architecture based on 6LoWPAN technology. Our initial numerical results show a reduction of the handoff costs on the mobile router which normally constitute a bottleneck in such a system. We also discuss important metrics such as signaling overload, bandwidth efficiency and power consumption and how they can be optimized through the mobility management.