Pictogram (象形) — two figures face-to-face, fighting with bare hands. Some oracle script characters of 鬥 show that both of their hair are bristled. The character shape resembles 𠨭 and 丮 opposite each other.
鬥 and 鬭 are original characters of this form; forms with the similar-looking radical 門 ("door") were non-standard variants.
Simplified form 斗 was derived via the variant form 鬦.
鬥 is also one of the 324 simplified characters introduced by the Republic of China (Kuomintang government) in 1935 as the simplified form of 鬭.
Conflation of two roots "to fight" and "to meet", and the merger has been attested since ancient times.
Probably from Proto-Sino-Tibetan*daw(“to defy, interfere, be at enmity with”), which is the source of Tibetanསྡོ(sdo, “to bear up against, bid defiance”) and Burmeseတော့(tau., “to resent an insinuation, interfere in a quarrel”).
Schuessler (2007) considers this cognate with 注 (OC *tjos, “to apply; to be touched; to bring together; to join”); see there for more.