Oh man, can I answer this one :)

This can really only be answered in opinion I suppose (although Im sure there is some "official" way in Academia- although I doubt anyone has the formula for a definitive answer or why would we study history at all?), and that the answer can be adjusted or personalized depending on your field- but for what it's worth, as someone who's nerd-ery is nothing but contradicting sources, this is how I dig as close to the truth as possible. A caveat: Every single one of these can be countered, even contradicted, by the other. :) I'll also try and give an example of how I use this at the end.

FIRST: Consider co-sources, surrounding sources, alternate witnesses

Seems the most obvious. Look at every source you can talking about the specific contradiction and see what the general trend is. Very often, we can make a safe assumption that the truth lies in what the majority witness. There will always be outliers, offshoots, variations- but much like mathematical data, the trend will be your safest bet. Within that, there will be a million details to work out, but you'll at least be able to get the general structure before working out the fine details.

SECOND: Consider the source it's coming from

Not all sources are equal. Source A might have been closer, but Source B was more involved. Source C might have been an effected party but Source D is able to look at the even with an outside eye. It's not that they aren't all valid- it's just that you're looking at different versions of the same story. Decide what exactly it is you're looking for. Strict facts? You'll have to strategically knock out opinion. Rippling effects of the event? You'll need to hone in on peoples reactions- including being able to pinpoint possible hyperbole, emotion, objectiveness, and investment. You'll then have to use all your research from Part One to try and shape the event while carefully cutting the fat away from lens of witness fallibility. Which leads to-

THIRD: Recognize your source

Your source is a real human, living during a real time, and their testimony is shaped by that. There are feelings involved, politics, social pressure, social norms. These people are subject to trauma, PTSD, survivors guilt, winners glory, prejudice, etc etc etc etc. It's all valuable but, in my experience, at least, official history can be distorted wildly by not understand the world in which the event took place lives in.

For a very brief example- my field is the sinking of Titanic. This topic is absolutely riddled with over a century of pop culture, spawning from a society foreign to those who research, but necessary to understand, truthfully the event. There are too many examples to give, so I'll stick with one that interests me quite a bit.

It is rumored, and scoffed at, that William Murdoch committed suicide shortly before Titanic went under. Historians can get very one sided on this (along with lots of other things), and it can make it hard to get down to the history after wading through the mess that is "the history" (if that makes sense).

In my opinion, after following (or trying to) the outline above- there can't really be any doubt that William Murdoch was the victim of suicide- and the answer to this is two fold. One- By painstakingly going through every source, every witness, placing them, timing them, weighing their legitimacy- the balance tips quite strongly in favor of suicide. When we follow that up with research into why the testimony is contradictory, we begin to understand why it's controversial. Then we start to understand why primary sources have, knowingly or unknowingly, contradicted their own testimony. Suddenly we start to see why James Cameron had to apologize for his movie scene, and why Will Murdoch comes up on reddit quite a bit, and why people debate this- and that, to me, is the real history.

That's a broad overview (that topic is a WHOLE other post unto itself), but it's a great example of comparing, contrasting, trimming the fat, having the evidence to justify trimming that fat, deciding what story you're trying to tell and saying, honestly, that "with all my best research- this is the series of events I believe is close to the truth as we are going to get'. Which, I suppose, leads to step four-

FOURTH: Some things we'll never know and will debate forever

And we'll just have to be at peace with that :)

Not sure if mods will keep this post as it's mostly opinion and personal experience. If not, totally fine- them's the rules- but I thought I could give an idea :)