Can I make a case for authorship if I spend two weeks running a series of repetitive experiments for my PI? I am a postbac working at a national lab. I have a project that is funded through an outside source, but my PI and postdoc informed me that they need me to spend the next two weeks completing a series of repetitive experiments they don't have the time to do for an alternative project. These experiments are required to satisfy the peer review criteria of the paper. Can I make a case for myself to attain authorship within the paper?
The short answer: 
It does not hurt to ask, and a good mentor or PI will explain the criteria for authorship and their reasoning for why you have/have not earned authorship.  It is a teachable moment.

The longer answer: 
Different journals give different criteria for authorship, and this is usually explicitly stated in the guidelines for authors.  In your question you mention that the experiments need to be performed to satisfy peer review.  If the article has already been reviewed for a particular journal (and perhaps now it is in a revision stage) then you and your advisor know which guidelines to check for their description of authorship criteria.  That said, it is a common criteria to require intellectual contribution (rather than simply data). If that is the case, and if you are very interested in this work, you could offer to help edit the paper.  However, if the paper has already been submitted, it can be challenging to add an author (for some journals this requires all other co-authors to sign off on the new addition).  If the paper has not yet been submitted, and the target journal is undecided, you really should discuss this with your advisor (which you should do in any case).  Personally I have worked for months on projects where I was not a co-author on the write-up when I was paid to primarily collect data.