It's a Guy Thing
As with the portraits of women with large coral rosaries, some of these portraits of men are clearly the "other half" of a pair of marriage portraits, while other portraits stand alone.
Besides rosaries, other favorite things for men to hold in formal portraits are a pair of gloves, a book, a skull(!) or a rolled-up piece of paper (though we usually can't see what's on it). Women tend to hold gloves as well, or handkerchiefs, but more often just have their hands neatly folded. (These are ladies of leisure, one presumes.)
Here are a few of the better portraits of Men With Rosaries that I've found, followed by links to several more.
Below is a portrait by Albrecht Dürer of his father (left), (1490), and a portrait of Lodewijk [Ludwig] van Gruuthuse, by the Master of the Princes' Portraits (right):
Below left, an unknown man with a rosary, by Bartholomäus Bruyn the Elder (mid-1500s); right, A portrait of a man from the Pilgrum family, by Bartholomäus Bruyn the Elder (1528):
Speaking of BIG beads, here's a portrait of a man from the Von Rhein family and a closeup view of his beads:
A portrait of a man from the Chemnitz family, by Ludget tom Ring the Younger (1569), and an anonymous portrait by the Master of the Bartholomew Altar (1492):
This one's rather entertaining: Stefan Praun poses for his portrait (painter & date unknown) dressed up in pilgrim's clothing. There's a large rosary hanging from the hand that's holding his pilgrim's staff.
Some people just don't like having their portraits painted. Perhaps the gentleman below thinks his beads are too small? (This is actually Duke Sigismund of Austria, Count of Tyrol, painted by the Master of the Mornauer Portraits, around 1470.)
See the links below for more portraits:
Otto von Langenfeldt, by a painter signing himself "M S V" (1510)
Man with Rosary, by Ulrich Apt the Elder, Augsburg (1st half of 1500s)
Portrait of a Man, by Bartholomäus Bruyn the Elder (1533)
A portrait by the Master of St. Sebastian of Mainz
by Bartholomäus Bruyn the Elder (circa 1528)
For more on men with big rosaries, see the entries for Counting to ten and Praying on all cylinders. For big rosaries held by women, see Big, Red, and German and Big Berthas.