Does the information relate to your topic or answer your question?
This particular question might not be as relevant (ha - see what I did there?) to your search for arguments because you're looking for arguments, not looking for sources for an essay. But remember, these strategies can be used in all your research, so it's important to include it here.
Who is the intended audience?
Arguments can be directed to specific audiences. So it may be important in your analysis to understand if a source has been written for a particular group of people. Look for clues in the article (or website) to determine if the author is addressing some defined audience.
Is the information at an appropriate level (i.e. not too elementary or advanced for your needs)?
Again, less important for your "argument hunt", but still important. You don't want to present an argument that is written for third-graders, right? (See point number 2 above!)
Have you looked at a variety of sources before determining this is one you will use?
Ask yourself, ia this particular source compelling? Is its argument compelling? (Yes, an article can be compelling emotionally while the argument is not sound, don't forget that!) Choose solid, well thought-out sources, and you will likely have stronger arguments to evaluate.
Would you be comfortable using this source for a research paper?
Or, would you be comfortable using this source in your argument journal? A spurious source may be a terrible choice for an argumentative paper, but if the argument(s) within are appropriate, even "ridiculous" sources may have value in this case.
Just remember, there's a difference between what you're looking for in this Logic class and what you might do for an English class!