With little searching I found that there is a recent book of translations: Ibn Sina's Remarks and Admonitions: Physics and Metaphysics An Analysis and Annotated Translation by Shams C. Inati that contains a specific reference for the name in Avicenna's writings, through Google books on p. 28 Inati writes:
...This proof is advanced by "the truthful", by which Ibn Sina seems to
mean "the philosophers."
Inati is referring to Part three: Metaphysics, Fourth Class, Chapter 29 pp. 130-131, so that is where you probably will find Avicenna's reference to the truthful.
EDIT: Through Google books again, yes it's there, and the meaning is explained there also, see also what Inati says on p. 28.
Searching "Burhan al‐Siddiqin" in Philosopher's Index I found the article "The Proof of the Veracious: An Exposition Based on ‘Allamah Tabataba’i’s Nihayatal-Hikmah" by Louis Medoff that contains the following paragraph:
The philosophers have offered various reasons why the proof is
attributed to the veracious. Siddiq means one who is extremely
truthful and not tainted by any falsehood. This truthfulness can
either manifest itself in act, meaning that the person’s acts are
purely for God without any showing off for others (riya’), or, as in
this case, truthfulness can manifest itself in thought, in so far as
the person’s intellectual path to God is not ‘tainted’ by relying on
others. As Sabziwari puts it in his annotations (hawashi) on the
Asfar, those who do not hold onto the reality of existence, which is the most clear and manifest reality as well as the most primary and
evident concept, and hold onto other things (such as coming into
being, contingency, or motion) will not be free from deviation and
missing the mark regarding the truth.