The top part looks like a 田 (rice field). Tatami mats are made from rice straw. The bottom part resembles 登 (to climb/ascend), which can remind you of stepping up onto a raised tatami floor. Envision the straw from the field being harvested, then neatly folded and piled to create these distinctive mats.
'Tatami' (for the mat) and 'Tatamu' (to fold). Think of 'TA-TA-TOO' many mats to fold.