The only real role of Víđarr in the norse mythology seems to be to avenge his father Odin during Ragnarokr by slaughtering Fenrir. In a footnote to his article Le dieu scandinave Víđarr , Georges Dumézil hints that it may be one explanation for his silence:
On a proposé d'autres explications pour
le silence de Vidarr; ce serait, par exemple, une prescription initiatique, Vidarr ayant besoin d'une «initiation» pour son exploit.
which translates roughly as: "Other explanations for the silence of Vidarr have been proposed: it could, for instance, be an initiation requirement, Vidarr needing an initiation in preparation for his achievement."
Interestingly, another story in which Víđarr can be found is the Lokasenna (Loki's Wrangling). This story relates a banquet that many gods attend (including Víđarr), the banquet is eventually interrupted by Loki and the poem turns basically into an insult contest between the gods attending the banquet and Loki. With one notable exception: Víđarr, who says nothing, but stands up to make a space for Loki (at the request of his father), and pour him a drink.
It may be possible that Víđarr is then referred to as "the silent one" for not taking part in that verbal joust.