Booyakasha comes from the Irish word "buíochas" (pronounced bwee ah kuss) which mean "glory to" "or praise be" as in "buíochas le Dia" which means "thanks be to god". Oliver Cromwell sent 1000s of Irish to the plantations in Jamaica, and a lot of Irish (or Gaelic as non Irish people know it) words made it in to Jamaican patois. Another example is "Irie" from the Irish verb "Eirigh" (pronounced "Ire - ee") meaning to get up.
Booyakasha is a polite way of saying booya.
It comes from hebrew בְּבַקָּשָׁה (b'vakashá) which means "please" or "you're welcome" and from booya (see booya)
Sasha Baron Cohen probably invented it for his character Ali G.