Conventional endovascular embolization of intracranial (or brain) aneurysms using helical detachable platinum coils can be time-consuming and occasionally requires retreatment due to incomplete coil packing. These shortcomings create a need for new biomedical devices and methods of achieving brain aneurysm occlusion. This paper presents a biocompatible and highly porous shape memory polymer (SMP) material with potential applications in the development of novel endovascular devices for treating complex intracranial aneurysms. The novel highly porous polyurethane SMP is synthesized as an open cell foam material with a glass transition temperature (Tg) of 39 °C using a sugar particle leaching method. Once heated above the Tg, the compressed SMP foam is able to quickly return to its original shape. An electrical resistance heating method is also employed to demonstrate a potential triggering design for the shape recovery process in future medical applications. The mechanical properties of the developed SMP foam are characterized at temperatures up to 10 °C above the respective Tg. The results from this work demonstrate that the porous SMP material developed in this study and the electrical resistance heating trigger mechanism provide a solid foundation for future design of biomedical devices to enhance the long-term therapeutic outcomes of endovascular intracranial aneurysm treatments.