By "choose a side" I really mean I try to weigh all the evidence for a side as best as time and ability allow

(I'm going to run with the hypothetical where someone meant this literally, even if you didn't.)

This should be "evidence against the side", if the choice is between primarily "for" and "against". Once you've made a tentative decision, additional evidence selected to support that decision won't change it, and so it does no useful work. There's also confirmation bias working in this direction. If on the other hand you focus on looking primarily for the opposing evidence, you may oscillate between various positions too much, but at least you'd be learning something in the process.

There is also a mode of gathering evidence where you improve understanding of the arguments already used to form your position. This understanding doesn't necessarily come with claims about how it'd sway your conclusions, it's motivated by value of information. The process of examining confirming arguments may look like gathering of more confirmating evidence, even if the outcome may be the opposite.