Do measurement protocols belong anywhere in the scientific publications? My research as a starting PhD student currently consists of reproducing and building upon the experiments carried out by other research groups 5--10 years ago. To this end, I have developed a number of measurement protocols, that is, step-by-step instructions how to clean samples, how to carry out microscopy measurements, how to troubleshoot experimental errors. 

I haven't seen this kind of information published or linked in scientific publications. Even the supporting information typically has a couple of graphs, a more verbose explanation of the materials and methods part, sometimes, a dataset, but nowhere close to step-by-step instructions. However, I feel that having this kind of information available will be a good help to other researchers, maybe to new researchers, and will serve to the reproducibility of the experiments.

Do these measurement protocols belong anywhere in the realm of academic publications? I have thought of publishing them in the online supporting information for the article, or in my PhD thesis, possibly as an appendix, or in my academic blog; which of those is popular / recommended way of publishing those protocols?

Research field is Physics / Material Science, if it matters.
There are a bunch of journals coming out for methods now, e.g. Methods in ecology and evolution.  Methods work is increasingly seen as a contribution in its own right.

On the other hand, it sounds like you may be writing quite an excellent review paper, and in that case you might include (unusually!) a methods section in your review article, and mention your methodology as one of the contributions in the article's abstract, intro, conclusion etc.

You should regardless of whether you publish the method be sure to include it as an appendix or part of the methods sections on your papers you use it on.  All papers need to be reproducable.  If you haven't published the paper featuring it yet, you will need to describe the full method in the first paper that uses it.  The editor will tell you whether to put this in electronic, appendix, or methods, don't worry too much on submission.  If you have already published it, you can just cite that publication and any differences / improvements since the publication.

Go ahead and blog it too.  Different people read blogs from papers.  Some people prefer to wait until they've got it published and past peer review before you go public on a blog.  But unless I thought the method itself was going to get me into a journal with an embargo (e.g. Nature / Science – unlikely!) I would probably blog it now.  I'd do that with the caveat on the top of the blog post in bold that it hasn't been published yet, and then another caveat at the bottom of the post in italics asking for feedback.  When your papers get published, be sure to go back and edit the blog post, edit out the caveats, and link to the new paper(s) so people who find your blogpost will cite your papers.