Is there any truth to the effect of repeating the name "Ar-Rahman" a certain number of times for strengthening memory, and does this contradict Islamic Sharia, which emphasizes memorizing the Quran for increasing memory and organizing life?

The act of worshipping by repeating Allah's name in isolation, such as "Allah, Allah," or "Hu, Hu," is an innovated Bid'ah. It has not been mentioned in the Quran, nor the Sunnah, nor by the predecessors (Salaf) of the Ummah, nor by its prominent figures whose example is followed. Rather, it was practiced by some of the misguided later generations. And the feeling of experiencing an effect from this does not necessitate its legitimacy.