A young Hans Willem Bentinck (l) and William, Prince of Orange (r). Detail from a portrait depicting William and Bentinck with a black servant boy and a white female figure, possibly serving as a personification of the province of Utrecht or the Netherlands in general, detail. Anonymous, c. mid-1670s, oil on canvas. Via RKD Images.
For all interested, here's a digitised copy of the letter in the Dutch Nationaalarchief.
The address "Mon Cousin" used by a person of a significantly higher to refer to a person of a significantly lower social status is no small thing within the strictly hierarchical and formulaic codes of conduct of the time.
Among the ruling families in Europe at the time, "Mon Cousin" would have been an appropriate address for either a monarch of equal rank, highlighting personal respect and regard for the addressee, or (or rather, often and) a close blood-relation of a (near-)equal rank.
To contextualise the significance: I was previously lucky to see a letter written by William III in his own hand to his cousin, the Electoral Prince Friedrich of Brandenburg (later Friedrich I King in Prussia and grandfather to Friedrich II "the Great") in an archive. In this letter, Friedrich is "Monsieur" to William, and only as he closes the letter does William speak of himself as "your cousin G[uillaume] Prince d'Orange" - which is essentially merely stating a biological fact - Friedrich and William were first cousins through the siblings Luise Henriette and William II of Orange.
Here, writing to Bentinck, William directly addresses his friend as "Mon Cousin", and puts further emphasis on the phrase and the meaning it entails by repeating it in closing the letter, which he signs with "Guillaume" only, utilising no further titles - which is striking, taking into account that the letter dates to early 1690, meaning William had very recently also taken the title of King of England as well as that of Stadtholder of the United Provinces. It is clearly William's intention to convey his trust and affection to Bentinck, and highlight the close personal connection between (not-so-) plain "Guillaume" and the man he professes to keep in the same regard as if he were a close blood relation.
Again, this expression is no empty phrase to write from William's perspctive, as he overtly likens Bentinck to being equal to him and the Orange-Nassau and Stuart families - a minor noble from Gelderland, a penniless third son equal to a Prince who had recently aquired the title of a King.
Naturally, given the formulaic, hierarchical social expectations and rules of the day, Bentinck could not have reversed positions and called William his cousin in a letter - though their letter exchanges suggest that in private at least, there may have been an informal tone adopted between the two.
In the past, there have been ample speculations whether the two may have been lovers, which is highly unlikely (the rumour was originally started by Liselotte von der Pfalz and is easily refuted by a wealth of contemporary documents, letter exchanges between William and Bentinck among them), but this letter really helps to understand the nature of their interpersonal relationship - William thought of them as as good as being a favourite relation.
This may be pure conjecture, but it seems likely that Bentinck, a middle child in a large family who all his life was particularly close to his big sister Eleonora Sophie, yet also had several younger siblings, felt instantly at ease adopting a similar role in William's life, who in turn had rarely had had contact to children of a like age growing up, and only met his only (known) illegitimate half-brother in his late 20s.
While we, people of a vastly different time (and in many cases, place) who have been socialised in vastly different societies, may not have the perfect words to express the nature of their relationship to exactness, their close bond, and the lifelong duration of their friendship (despite some troubles in later years) is rather touching.
And because the only thing that is ever cited is how Beninck stayed with William during the nights of his smallpox-infection, here are a few examples of how William in turn supported his friend:
In 1675, Bentinck was head over heels in love with Charlotte Philliberthe von Nassau-Beverweerd, a distant cousin of William's from a minor branch of the Nassau family, and seems to have proposed to her. "Lotte" as she was known to William, promised she would give Bentinck an answer - and then ignored both the patiently waiting Bentinck and the a lot less patient William, who tried to get Lotte to give her answer (whichever one it would be) by letter. For the full details, see my post here.
In February 1678, Bentinck married Anne Villiers, an attendant of William's new child mail order bride Mary Stuart. Anne had been set on a life as an unmarried lady-in-waiting in Mary's household as her family did not have the means to pay dowries for her and her sisters. Combined with the fact that the future Mr and Mrs Bentinck only had effectively known each other for approximately 3 (!) moths, theirs seems to have been a love match. To save the bride some blushes on the financial front, William awarded her a dowry from his private fortune.
William was godfather to 2 generations of young Bentincks, and sometimes, he would receive visits from them, unaccompanied. As one remarkable letter indicates, William made a point of being informed on Bentinck's parenting, and respected his rules - such as not allowing the teenage Henry Bentinck to go on hunting parties, no matter how much pleading and sulking would be involved.
And here is William's wish for Bentinck on the birth of his son Willem (called Willemtie by the family, including William) in 1681: "[...] who I hope will live long enough to become as honest a man as you are. And if I should have children, I hope they will love each other as much as we do."