The poem "Mr. Flood's Party" by American poet Edwin Arlington Robinson, is a sad tale of the closing phase of one man's life. He has lost many friends over the years since they died before him: "Where friends of other days ...." He goes to a hilltop above the town, under "the harvest moon" of autumn.
It is a place that he has frequented in the past, a place of seemingly fond memory, but a place he hasn't visited for a long while: ""Well, Mr. Flood, we have not met like this / In a long time; ...." While on the hilltop, he talks to himself, calling himself "Mr. Flood" as though he were a separate person, and toasts the friends of the past while he sings "For auld lang syne" in tribute to their memory. It is with this understanding of the poem that you can identify the meaning of lines 23 and 24:
A phantom salutation of the dead
Rang thinly till old Eben's eyes were dim.
In this context, "phantom salutation" is the eerie greeting sent him through his memories of the friends who are no more: the whispers of greeting that “Rang thinly” from long dead friends and memories of times together. So "salutation" here means greeting. This understanding is confirmed in the four final sad lines of the poem:
There was not much that was ahead of him,
And there was nothing in the town below—
Where strangers would have shut the many doors
That many friends had opened long ago.
Many years ago, friends had opened the doors of the cottages in the town below the hilltop, now strangers close the doors to houses where no friend dwells within. There was not much remaining of his life, in the autumn of it's days, ahead of him and there was nothing of friendship awaiting him in the sleeping town below.