The Cambridge Bible for Schools and Colleges is especially illuminating here,
It may have been the
custom, upon occasion of a death, to offer some prayer or invocation
to Jehovah; and the speaker, unmanned by the terrible mortality about
him, feels a superstitious dread of mentioning Jehovah’s name, lest He
should be moved by it to manifest some fresh token of His displeasure
(comp. partly Isaiah 19:17).
Invoking god's name upon the dead was common practice in ancient Israel, the relative and burner knows what's coming and he quickly hushes him from fear of mentioning the Lord's name. If this interpretation is correct, then we have here the earliest reference for the Jewish Mourner's Kaddish in the bible itself (though the subject of the Kaddish is not death, the prayer is nevertheless associated with death).
The IVP Bible Commentary concurs with the idea that it is the fear of invoking his name that brings him to hush,
not mentioning the Lord’s name. God’s wrath is such that the
population of the city of Samaria is to be reduced to a tenth, and the
survivors will be so frightened by what Yahweh has done that they will
be afraid to mention God’s name lest the angry deity take any further
notice of them. In that sense then, the command “Hush!” is a sort of
warding spell (like “God forbid” in English) to prevent the incautious
from invoking God (compare Ex 23:13 and Josh 23:7). Assyrian royal
documents from the reign of Enlil-Nirari (1326-1317) provide some
light here. In one text the king calls out, “May the god by no means
speak!” when the death of a member of the royal family is announced at
court. His intent may be to ask that the god not act (speak) against
anyone else.
Alternatively, it is not the fear of mentioning god's name that overcomes him. It is rather the stubbornness to recognize the hand of Yahweh that makes his superior interrupt him from mentioning god's name. Amos is here picturing a scene of where the people are in total despair and there is dead people in every house, but the people are still determined to continue in their old corrupt ways.
Here is a similar explanation from Barnes' Notes on the Bible,
The words then, "for not to be mentioned is the Name of the Lord," are
very probably the voice of despair. "It is useless to name Him now. We
did not name His Name in life. It is not for "us" to name it now, in
death."
