|Kanji in this term|
Etymology 1 
The original kun'yomi pronunciation of this word was ashibi, now considered a variant reading.
The leaves of the plant were known to intoxicate horses when eaten, causing horses to stumble and become weak in the legs. As such, this word is most likely a contraction of 足癈 (ashijihi, “numbness in the feet/legs”).
The kanji 馬酔木 are an example of jukujikun, meaning literally horse intoxicating tree.
See also 
Etymology 2 
The on'yomi of the kanji spelling 馬酔木.