The 21-year-old, a former Wales Under-20 international, made his PRO14 debut for the Blues earlier this season.
The Blues announced deals for back-row forward Botham, whose father Liam played for Cardiff, prop Keiron Assiratti and flank Shane Lewis-Hughes.
"It's great to sign my first professional deal, and I am really happy it's at Cardiff Blues, where I've spent the last three years in the academy," Botham said.
"You only have to look at the likes of Owen Lane, Jarrod Evans and Dillon Lewis to see how quickly young players can progress at Cardiff Blues, so this is definitely the right environment for me to improve as a player," he added.
Botham was born in Cardiff and he has also represented Wales in sevens and at under-18 level.
Botham's grandfather, Ian, 63, is considered one of the finest all-rounders in cricket history and was one of England's top sports personalities in the 1980s.