The data writing and thermal stability of information storage are studied theoretically for a magnetic random access memory (MRAM) composed of a magnetic tunnel junction or multilayer exhibiting giant magnetoresistance. The theoretical analysis focuses on the magnetization switching in the free layer of a MRAM cell, which is induced by a spin-polarized current imposing a spin-transfer torque (STT) on the magnetization. It is shown that the writing current in such an STT-MRAM reduces dramatically near a spin reorientation transition (SRT) driven by lattice strains and/or surface magnetic anisotropy and even tends to zero under certain conditions. In particular, at the size-driven SRT in the perpendicular-anisotropy CoFeB-MgO tunnel junctions, the critical current densities for magnetization reorientations between the parallel and antiparallel states are expected to fall to low values of about 1.3 x 105 and -3.3 x 104 A cm-2. Remarkably, STT-MRAMs may combine low writing current with very high thermal stability of information storage (retention over 10 years) even at a high density approximate to 500 Gbit inch-2.