As you've succesfully guessed, it's not tied to a technology but to a quest (Deeds rewards are given via hidden quests), if you're just trying to make it available to everyone here's a very simple change :

In \Endless Space 2\Public\Simulation\Battles\ModuleDefinitions[DLC1].xml

Delete this line :

<PathPrerequisite Flags="Edition" Inverted="false">../ClassEmpire/ClassQuests,CompletedQuestMilitary_Stage4A</PathPrerequisite>

So I made a new tech, with a new SimulationDescriptor entry, copied the Annihilator Beam code from ModuleDefinitions[DLC1].xml into a new module XML, removed the line you quoted me, added a line so it had a tech requirement consisting of the new tech I made, and added the tech unlock code to my faction trait XML. Still doesn't show up in game... Annihilator Beam can go on small ships right?

Is there a step by step somewhere on how to make a new module? I can make a trait that grants the Sophon Free Move module just fine, but I can not for the life of me get a custom module to show up in game, even if I don't give it any prerequisites.

I know it's asking a lot, but could someone look at my files? I've got code for traits and technologies and such in there, but right now I'm just trying to figure out why I can't make a new module...

HiddenTechs.zip

Although ideally I would like to figure out how to just make a trait that unlocks existing modules directly, but I don't think that's possible without them being tied to a technology, and I don't think you can make a technology prerequisite for an existing module without breaking the normal method of obtaining it (like a Deed).

It looks like you're missing extratypes to load stuff in your mod file. The main one I noticed was lacking was the WeaponModule one.

Check out this link for a complete list

Edit: Oops, looks like you had that commented out, my IDE just hid the comment. Try making a non-weapon module, I've had problems with weapon modules too.

I tried to do a new module more or less from scratch, simple as I could; I just made a clone of the Sophon Free Move and didn't add any prerequisites. A module without prerequisites is available by default right?

Here's the file if anyone feels like giving it a look.

ExTech.zip

EDIT: Forgot to mention, it doesn't work