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In the front matter of some books, before the half-title, on the very first page, is a small excerpt from the book, sometimes slightly edited in relation to the actual content, as a kind of teaser. In a paperback, it's normally on the first recto immediately after the cover.

Does this excerpt have a specific name?

A number of people involved in publishing on demand or Amazon.com marketing use the term front matter excerpt:

Scott Merrill will record a special scene — perhaps the cover scene — of your latest book, or the back cover blurb and front matter excerpt and review snips, or a short (2-4 minutes long) pitch for your work. — Brenda Novak’s On-line Auction for Diabetes Research

Prepare front matter excerpt if desired, copyright page, and any additional material, i.e. About the Author, Glossary, Author's Note, etc. — Ebook Conversion Tips, Nancy's Notes From Florida (blog), 27 July 2011.

The Front Matter excerpt (and "See a random page") are based on the PDF file uploaded to the old Look Inside program. — Forum post, KBoards.com, 26 July 2012.

I had a good time taking the paperback of Worldwar: In the Balance around to my friends and getting them to read the front-matter excerpt. NielsenHayden.com (blog), 22 Sept. 2007.

Although I wouldn’t consider any of these sources authoritative, the term is readily understood by anyone who knows what the front matter of a book entails. I have only seen front matter excerpts in works of popular fiction in paperback, especially by relatively new authors.

A professional editor discusses a fairly exhaustive list of what components constitute front matter:

Frontmatter

Anything that comes before the start of the text is considered frontmatter. Usually, frontmatter is numbered using roman numerals, and the text (or half title page, or part opener) starts with page 1. Here are some things you might see in the frontmatter (some of these are required, and we have to make space for them no matter what):
Teaser/Excerpt page (optional) — you may decide you want to put in a short, compelling teaser of the text as the first page of a book, to give readers a taste of the book. — Christine Barcellona, “Ask an Editor: What are the parts of a book and how are they laid out?” Swoonreads.com, 22 Feb. 2016.

Other terms in the list are “praise page,” a page of favorable reviews, and “ad card,” a list of books by the same author (or, I presume, the same series.) She also mentions that an excerpt can be included in the back matter, say, of the next book/volume in a series or for another title that might interest the reader.

I know no standard name for this feature, so I would default to a name that describes the content. (Nota bene: I have mainly worked with medieval manuscripts and early modern books.)

Because the half-title or bastard-title is so often the first recto of a volume, what precedes the half-title has no formal name. If it's a listing of review blurbs, it can be called a number of names, including "Endorsement," "Advance Praise," or "Praise." If the text is a blurb when it's on a back cover, it's also a "Blurb" on the front cover. Following the same principle, an excerpt from the text that's being used as a preview or promotional hook can be called a "Preview" or "Hook".

I've seen this appear on the front endpaper (flyleaf/pastedown) or on the first page preceding a half-title or full title.

It's not an epigraph, since an epigraph customarily quotes another text, and is usually an exercise in comparison or juxtaposition: how does the outside text flavor the way the text will be read?

I would argue it is called an

abstract

An abstract is a self-contained, short, and powerful statement that describes a larger work. Components vary according to discipline. An abstract of a social science or scientific work may contain the scope, purpose, results, and contents of the work. An abstract of a humanities work may contain the thesis, background, and conclusion of the larger work. An abstract is not a review, nor does it evaluate the work being abstracted. While it contains key words found in the larger work, the abstract is an original document rather than an excerpted passage.

This is according to https://writingcenter.unc.edu/tips-and-tools/abstracts/.

It could also be called a blurb.

Blurb

A short description of a book, film, or other work, written and used for promotional purposes.
This is according to https://en.wiktionary.org/wiki/blurb