Generate and save to file a random string at compile time

As far as I can see, your problem can be broken down into two parts

1. Storing and retrieving the random string in your code
2. Generating the random string at build-time

Storing and retrieving the random string in your code

This is the easy part. Because it will be auto-generated each time, you need to store it outside of your code. You can store it in a database table, a file, windows registry, on a remote server - it's up to you. That way, your code only needs a function to retrieve it from the location you saved it.

Generating the random string at build-time

I assume you are using visual studio, from the tag in the question. Here you can use the power of MSBuild. You would need to edit your CSPROJ file, to add the action to generate your random string during build and to push the generated string to the store location you chose above, inside an MSBuild targer.

The MSBuild targets you are interested in are either of these two:

<Target Name="BeforeBuild"><!--Runs before build begins-->
</Target>

And

<Target Name="AfterBuild"><!--Runs after build completes-->
</Target>

You can run literally anything within these targets. For an example helloworld using C#, take a look at the following link MSBuild Hello World on the Microsoft Documentation site.

Side note: I recommend the book Inside the Microsoft Build Engine - Using MSBuild and Team Foundation Build by Sayed Ibrahim Hashimi and William Bartholomew.