Bookmarks are cheap enough that nobody specifies them, which is exactly why most of them are bad. Four questions fix it.
Caliper, in microns
Ask for a number. Premium card means nothing; a caliper in microns can be compared between two quotes and matched to the books it will live in.
Lighter board for paperbacks, heavier for hardbacks. A single spec across both is a compromise for one of them, and usually it is the paperback that suffers.
Laminate or uncoated, decided by whether people write on them
Laminated survives a bag far better and is noticeably slippery, so it slides out of a book more easily — which is fine for a giveaway and annoying for a reader.
Uncoated is warmer in the hand and takes pencil and pen. If the bookmark is for a library, a reading group or a school, that usually settles it.
Rounded corners, always
A square corner on a card carried in a bag goes soft and dog-eared within a fortnight. Rounding costs essentially nothing and doubles how long the bookmark looks new.
It also stops the corner catching on pages as the bookmark is pushed into a closed book.
If it has a tassel, it needs an eyelet
A knot pulling on a bare punched hole tears card. It is not a question of if.
A small metal eyelet spreads that load into the board, reads as a quality signal, and adds very little to the unit price. Any tasselled bookmark quoted without one will shed its tassel.
What to send when you ask for a quote
Artwork, quantity, whether they are for paperbacks or hardbacks, and whether anybody needs to write on them. That is enough for a real quote and a mockup the same day.
For a shape, send an outline — and expect us to say so if a neck in it is too thin to survive being pulled out of a book.