The word "silver" appears in the sixth stanza, after Eben Flood has "returned" with himself and is sharing a drink, also with himself; it is hinted that he may be celebrating New Year's Eve, or perhaps his own birthday. The song of "Auld Lang Syne" would be appropriate for either situation. Flood is old and lonely, and creates for himself an alter ego to share his moment of joy, named "Mr. Flood." The "two" share a drink and a song, and the narrator speaks about Flood's song:
"Only a very little, Mr. Flood --
For auld lang syne. No more, sir; that will do."
So, for the time, apparently it did,
And Eben evidently thought so too;
For soon amid the silver loneliness
Of night he lifted up his voice and sang,
Secure, with only two moons listening,
Until the whole harmonious landscape rang--
(Robinson, "Mr. Flood's Party," poetryfoundation.org)
Flood is standing alone in the moonlight, which is often referred to as "silver," but the word does not refer to the moon (which Flood briefly sees doubled, due to his intoxication) but to Flood's own old age and hermit-like status. Flood is an outsider to the town below, where he was once a valued member of society, and he is past his "golden years." In fact, since the harvest moon is more-often referred to as yellow or gold, the reference to his "silver loneliness" is likely because of his age and the fact that all his friends and family are gone. In his "silver" old age (greying hair is often called silver), Flood has no one to share his time but himself; he is regretful of his choices, and perhaps of a wasted youth. Just as silver is seen as less-valuable than gold, Flood is less-valuable to society than other people, and so his loneliness is based in his isolation.