Offhand I can think of 3 main purposes for scientific journals: rapid dissemination, archiving, and evaluation (correctness and importance). The first ceased long ago to be feasible, and preprint servers and newer technology such as blogs have largely taken over. The second is still important, and a decent job is being done by some commercial (and other) publishers.

The third is perhaps the most important. The main questions, according to G. H. Hardy, that a referee should ask are: is it true? is it new? is it interesting? In my experience not many papers pass all three of these tests.

The first of Hardy’s questions can be hard to answer: from my own experience as author and referee I know that errors can easily slip through. In mathematics, there may be some hope of eventually automating proof-checking, or perhaps using the idea of zero-knowledge proofs. In other fields, I am not sure what more can be done. It would be very interesting to know how many experimental results are never checked by anyone other than the author(s). It has been asserted that most published results are wrong.

The second test is more easily answered now with the advent of full-text searching. Provided the archives are readily available, modern search technology makes it less defensible than ever not to be aware of related literature when publishing. Referees can of course use this technology too.

The third test is a matter of taste, and rejecting a paper on the grounds that it is boring or trivial requires more skill and judgment on the part of the referee. Many rejection letters say, in effect: this paper is OK, but not good enough for this journal, or not of interest to “our readership” (whoever they are).

I suggest that papers that pass two of Hardy’s three tests should usually be published, and new journals be instituted if required. I only discuss mathematics here.
Results that are false, but new and interesting should appear in the Journal of Speculative Mathematics. To some extent, the arXiv can fulfil this function, but papers there often fail more than one of the three tests.

Results that are true, already known, but interesting (with a new and clearer proof) should appear in textbooks or the Journal of Expository Mathematics. There are already several expository journals, of course, such as the American Mathematical Monthly.

Results that are true, new, but boring should appear in the Journal of Necessary Mathematics. The Journal of Necessary Mathematics would be better than the recently introduced journal Rejecta Mathematica (which seems to be having trouble attracting submissions). JNM would contain lemmas that are likely to, or have already proven to be, useful building blocks in more interesting results. Often they are easy to believe but hard to prove nicely. Currently these are often relegated to appendices of papers, and never read. It would be far better for them to be archived systematically. At the very least, they would provide a nice resource for people looking to practice their expository skills – try to find a better proof, or an application.