With the last update of the official BIPM brochure The International System of Units, 9th edition, 2019, the unfortunate translation "thin space" was removed. They also fixed the "une espace" issue mentioned as a comment.
This makes it very clear that the SI wants to prescribe the presence of spaces at certain positions, but not their exact width. The width of spaces should be up to the typografic expert. This is the same way, orthography rules for a natural language talk about spaces between words.
Answer as of 2016/2017
After research, I agree with Javier Bezos in that the BIPM does not say anything about the size of spaces.
Some argue that the BIPM says “The numerical value always precedes the unit, and a space is always used to separate the unit from the number.” and this is should be read as being a normal space, because at other places, a “thin space” is mentioned. I challenge that. Here is my argument.
My reference is the official BIPM brochure, SI, 8th edition, 2014. It contains the French and the English version.
Here are all mentions of typographic spaces (without the “no space here” ones).
Section 5.1 talks about spaces within units.
La multiplication doit être indiquée par un espace ou un point à mi-hauteur centré [...]
Multiplication must be indicated by a space or a half-high (centred) dot [...]
Section 5.2 talks about writing units in their long form (“pascal seconde”)
[...] il convient d’utiliser un espace ou un tiret pour séparer chaque nom d’unité.
[...] then either a space or a hyphen is used to separate the names of the individual units.
Section 5.3.3 is the famous quoted one.
La valeur numérique précède toujours l’unité et il y a toujours un espace entre le
nombre et l’unité. Ainsi la valeur d’une grandeur étant le produit d’un nombre par
une unité, l’espace est considéré comme un signe de multiplication (tout comme
l’espace entre les unités). Les seules exceptions [...]
Cette règle signifie que le symbole °C pour le degré Celsius est précédé d’un espace
pour exprimer la valeur de la température Celsius t.
The numerical value always precedes the unit, and a space is always used to separate
the unit from the number. Thus the value of the quantity is the product of the number
and the unit, the space being regarded as a multiplication sign (just as a space
between units implies multiplication). The only exceptions to this rule [...]
This rule means that the symbol °C for the degree Celsius is preceded by a space
when one expresses values of Celsius temperature t.
Même lorsque la valeur d’une grandeur est utilisée comme adjectif, il convient de
laisser un espace entre la valeur numérique et le symbole de l’unité. [...]
Even when the value of a quantity is used as an adjective, a space is left between the
numerical value and the unit symbol. [...]
Section 5.3.4 is interesting in that the English translation introduces a “thin space” that is not in the original French version. English and French texts differ.
[...] les nombres comportant un grand nombre de chiffres peuvent
être partagés en tranches de trois chiffres, séparées par un espace, afin de faciliter la
[...] for numbers with many digits the digits may be divided into groups
of three by a thin space, in order to facilitate reading.
Quand il est
utilisé, il convient de mettre un espace entre le nombre et le symbole %.
When it is used, a space separates the number and
the symbol %.
Very important is the statement about the English translation.
To make its work more widely accessible, the International
Committee for Weights and Measures has decided to publish an
English version of its reports. Readers should note that the
official record is always that of the French text. This must be
used when an authoritative reference is required or when there is
doubt about the interpretation of the text.
Reading this, the argument that “a mentioned ‘space’ must be the same width as a text space, because the text knows a ‘thin space’ at other places” is void, because there is no such thing as a “thin space” in the French authoritative reference.
(My further interpretation is that “espace”/“space” does not say anything about the space's width. It's the same way the English grammar talks about words (and spaces separating them), while the width of a normal space in body text is, firstly, up to the font designer and typographic user and, secondly, may stretch or shrink within a text to achieve a desired typographic layout like justified text.)