If you are asking about the meaning of alliteration in the phrase, alliteration is a sound device found in poetry and prose in which the beginning consonant sound of a word is repeated in other words in the same line or phrase. In the phrase you mention, the sound of "S" is repeated in the words "slumbering," "spark," and "sociability." The effect of the alliteration is to emphasize these words in relation to Ethan's personality. In his loveless marriage to Zeena, his life has become one of emotional withdrawal. The isolation of the winters in Starkfield has also cut him off from human companionship. However, there is something in Ethan, some "spark" that still longs for warmth and human contact. That spark is ignited when Mattie comes to his and Zeena's house.

With this alliteration, the author Edith Wharton is telling the reader something about Ethan's character.  On the surface he appears to be "grave and inarticulate", especially in the isolated circumstances of his living and marital relations. Ethan was at one time a student of engineering, but was forced by family situations to assume life on a small farm.  He is married to Zenobia, but she is bitter and a hypochondriac, and provides little positive interaction for him.

Despite all this, in the alliteration

"There was in him a slumbering spark of sociability",

the author is telling the reader that deep within Ethan's nature there is the inclination to seek "sociability", the give-and-take of true interaction with others.  It is true that this inclination is almost undetectable; due to a long period of suppression, it is not vibrant, but is "slumbering".  Like a fire that has been long deprived of oxygen, it is almost completely snuffed out - only a "spark" remains.  Nonetheless, it is there, dormant, only waiting for a breath of fresh air to bring it to life again, a breath that is provided in the character of Mattie.

By using the sybilant sound of "s" repeatedly in this alliteration, the writer conveys the sense of quiet dormancy.  The "slumbering spark of sociability" lies hidden beneath the surface, silent, but definitely present.