The Lakers announced the deal Friday night before their game against the Portland Trail Blazers.
Buycks played in 14 games with the Toronto Raptors last season, averaging 3.1 points and 1.6 rebounds.
He played in Spain and China this season before joining Oklahoma City in the NBA's D-League, averaging 18.2 points, 6.8 rebounds and 6.2 assists in 10 games to earn his call-up to the Lakers.
Buycks played two seasons at Marquette in his native Milwaukee before embarking on a pro career in 2011 that took him to Belgium, France and the D-League.
Earlier in the week, the Lakers signed D-League call-up Jabari Brown to a multiyear contract.