Pictogram (象形) — two figures face-to-face, fighting with bare hands. Some oracle script characters of 鬥/斗(dòu) show that both of their hair are bristled. The character shape resembles 𠨭 and 丮 opposite each other.
鬥/斗(dòu) and 鬭/斗(dòu) 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/斗(dòu) is also one of the 324 simplified characters introduced by the Republic of China (Kuomintang government) in 1935 as the simplified form of 鬭/斗(dòu).
Conflation of two roots "to fight" and "to meet", and the merger has been attested since ancient times.
"to fight"
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”).
"to meet"
Schuessler (2007) considers this cognate with 注 (OC *tjos, “to apply; to be touched; to bring together; to join”); see there for more.