Lanthanide-based single-ion magnets have attracted much interest due to their great potential for information storage at the level of one molecule. Among various strategies to enhance magnetization blocking in such complexes, the synthesis of axially symmetric compounds is regarded as the most promising. Here, we investigate theoretically the magnetization blocking of several lanthanide ions (Tb3+, Dy3+, Ho3+, Er3+, and Tm3+) encapsulated in highly symmetric zigzag boron nitride nanotubes (BNNTs) of different diameters with ab initio methodology. We found that Tb3+@(7,0)BNNT, Dy3+@(7,0)BNNT, and Tm3+@(5,0)BNNT are suitable SIM candidates, while the other investigated complexes from this series show no signs of magnetization blocking owing to a hard competition between contributions to the crystal field of the lanthanide ion from neighboring and more distant atoms of the nanotube.