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.
In addition to the answer provided by user2860703, you may also need to consider the policies of your university. In germany, almost all universities follow the recommendations of the DFG (German research agency) (pdf, german). It explicitly excludes "honorary authorships", and states as one of the reasons that do not qualify you as an author (page 30 of the linked PDF) "purely technical assistance in data collection" ("lediglich technische Mitwirkung bei der Datenerhebung").
To qualify (as per these rules), you would need to make "significant contributions to the concept, development, analysis / interpretation and writing of the paper" ("Konzeption der Studien oder Experimente, zur Erarbeitung, Analyse und Interpretation der Daten und zur Formulierung des Manuskripts"). Disregarding this policy is academic misconduct and may, in the worst case, lead to your degree being revoked.
I have not heard of a case where a degree was revoked because they gave someone a honorary authorship, but be aware that these rules exist if you work in germany, and that your institution may have similar or additional rules for this. Enforcement of these policies probably varies across institutions and countries.
This is just my 2 cents, but I think far too many of the answers here reflect "ideal science" rather than what actually happens. You don't get authorship on a paper because you met all the requirements for authorship that the journal asks! haha, gosh, no way. Here are what I suspect are the most important factors that contribute to authorship ranking (in Biology):
1) Person Directly Responsible for the project.
Is it a PhD project (where a PhD student is directly responsible), or it's a project being supervised by a post-doc (who is directly responsible) but work is being carried out by others? The PDR has the most right to authorship, regardless of how much time at the bench they put in. When i was a BSc, I did months of work for a post-doc, and of course I wasn't an author on anything, because it was his project not mine.
2) Person who wrote significant portions of the paper.
This is surprisingly the second most important factor, after the PDR, because it's very difficult to justify being higher-up on the authors list of a paper that someone else physically wrote. Often the PDR is also the person who writes the paper, so this is rarely an issue, but when thats not the case it's important to bare this in mind. There's a person in my lab who wrote up a paper using the results from a PhD student who after graduating (but before publishing) left for a post-doc. The ex-PhD student being the PDR is number 1 on the paper, and the person who wrote it up is number 2. Myself, a PhD student who contributed a key experiment (and considerable time) to the project, was somewhere in the middle, because I obviously have no leg to stand on demanding authorship on something i didn't write.
3) External lab wild-card bonus.
The more your PI wants to collaborate with an external lab that did something on a paper, the more likely that external lab will be placed high on the authors list. This is particularly true if the external lab is a medical facility which provided sample material. I've seen people who have supplied biopsies being credited as 1st author over the PhD students who worked for 3+ years on the project because the PI needs to maintain a good relationship with that medical facility if they are to expect more samples in the future. More often than not though, these people end up as 2nd/3rd authors with everyone else apart from the PDR and PI wondering who the hell they are.
4) Results contributed.
This is different to work, because results are measured in figures, not hours. Basically, if one of your experiments determined the take-home message of the paper, you contributed a result. If your work added support to the take-home message, that may or may not be a result, depending on the PI. Statisticians/Bioinformaticians can get lucky or unlucky here, depending on how well their PI understands their job. If you run data through a standard analysis pipeline and the results end up looking interesting, and this may give the bioinformatician more authorship credibility to some PIs than the PhD student who had to optimise the protocol to get that data in the first place (even if that optimization took months/year), because the output of protocol optimisation is what we expected to work now works, whereas the running of data through a pipeline lead to a new research direction.
As you can see, work comes pretty low on the list - in fact it takes considerable work to even make it on a paper if you have not contributed results. If you are proving something the PDR knows is likely to be true, as is the scenario in the OP, the chances of authorship are essentially up to the benevolence of the PI/PDR.
Finally, I don't agree with the system we have, obviously, because i've been stung in just about every way imaginable by it - but that's just how it is. Authorship ordering is not a science, it's political, but having been in Academia for a while now i'm slowly coming to appreciate why the system works the way it does - there are often just too many diverse stakeholders in a paper to apply any system other than a political one. There is no authorship formula.
These days, now that i've nearly finished my PhD, i've noticed that this system benefits me more than it used to. I've contributed in small but important ways to numerous projects, and as a result my authorship average is getting higher for less work. No doubt, any system that benefits those who know it the best, is a system that is likely to remain until it gets overthrown by the angry masses.