The phrase "to construct an argument" means to create or build a logical and convincing explanation or point of view using evidence, reasoning, and careful organization. Full definition
Our clients then construct the argument from the authorities that we provide. (onpointlaw.com)
That we win culture wars with carefully constructed arguments and political power plays, or that we win hearts with humility, truth, and love? (relevantmagazine.com)
Until the left recognises its mistakes it can not construct an argument at all and the problem is, that as so often, it is so deeply conservative. (nextleft.org)