What is Henry's brilliant idea with regard to Thomas Becket?Why does he do this?

Henry II, King of England, was very interested in maintaining and expanding the power of the monarchy.  He was especially concerned with making sure that the Church did not get to be too powerful and too independent.

In Act I of this play, Thomas Becket is Henry's friend and his chancellor -- the official in charge of collecting taxes.  Becket and Henry have trouble getting the Church to pay the taxes the King thinks they should.

Henry's brilliant idea is to appoint Becket as Archbishop of Canterbury (head of the Church in England).  That way he will be loyal to Henry and not to the church and Henry will have more power.

Becket does not side with Henry and Henry, of course, ends up having him killed.