When reviewing for MathSciNet, I routinely find myself just paraphrasing and abbreviating the introduction provided by the author, and occasionally adding a few words about the quality of the research or the cleverness of the argument (which the authors themselves would not be able to write for obvious reasons). There seems to be very little added value in doing this, since the paper already has the introduction (which has the added benefit of being written by someone who has intimate knowledge of the paper) as well as abstract (which will in most cases be sufficient to decide if the paper is worth reading). Of course, I can imagine edge cases when someone can't quite decide if the paper is worth delving into based on the abstract alone, while the introduction is for some reason difficult to read or the paper is difficult to access. But I can't help feeling that there should be more to it.
I would love to hear opinions about what makes a MathSciNet review useful, and how to achieve it.
Edit to add: As YCor correctly points out, the same question applies with ZBMath or any other place that hosts public reviews in place of MathSciNet. To avoid creating a question which is a moving target, I will refrain from making edits above.
Best Answer
I'll take a stab at this because in the past I have gotten some feedback from Mathematical Reviews saying that they like my reviews, and they did ask me to write a Featured Review once (back when there were such things as Featured Reviews).
The answer to the question depends to some extent on how long a review you want to write. My default length is probably around two to three times the length of the abstract. I typically try to at least give a precise statement of the main result(s). Often the abstract does not do this because stating the main result requires quite a bit of notation and preliminary definitions, which are too long to put in the abstract, but which usually can fit into a review. I do this because I imagine that some people might have access to MathSciNet but for some reason don't have access to the paper, and a precise theorem statement might help them decide whether to put in the extra effort to obtain the paper itself.
Another thing I do is to put myself in the shoes of a MathSciNet user, who is looking for relevant papers that he or she is not currently aware of. I ask myself, what keywords can I put into the review that will help such people discover this paper via a keyword search? If you look through the paper with this mindset, you will often find remarks about related topics that will provide good keywords. These don't always make it into the abstract or the introduction, but it's useful to have them in the review. If you happen to know that the objects in the paper are sometimes studied under a different name then that's also something useful to mention in the review.
If you're willing to put in the effort, MR will happily accept longer reviews. As I understand it, the Featured Reviews that MR used to have were discontinued for various reasons (e.g., I heard that, contrary to MR's intent, Featured Reviews were being used by the community for hiring and promotion decisions, and MR did not feel qualified to decide which papers were the "best"), but there is nothing to stop you from writing something similar for any paper you feel like. You can search for "featured review" in the review text of reviews from June 2005 or earlier to get a feeling for what these were. Well-written Featured Reviews were not only longer and more detailed than the typical review, they were written with a wide audience in mind. The idea was that a Featured Review would convey some idea of the context and significance of the paper to non-specialists. I will freely admit that I rarely have the energy to write such reviews, but they are certainly of value. Imagine someone stumbling upon your review in their search results and finding your review more accessible than the paper itself; they could very well make a conceptual connection that they wouldn't have otherwise, or be drawn into an area that is close to their own interests but that they didn't know existed.