The meaning of Dregs in the Bible
(From International Standard Bible Encyclopedia)

dregs: The "sediments," "lees," "grounds of liquor"; only in plural. In the King James Version it stands for: (1) Hebrew qubba'ath, "bowl," "chalice," found only in Isaiah 51:17, Isaiah 51:22 : "the dregs of the cup of trembling"; "the dregs of the cup of my fury." the Revised Version (British and American) correctly changes "dregs" into "bowl." (2) Hebrew shemarim, "sediments" or "dregs," especially lees of wine. "The dregs thereof, all the wicked of the earth shall wring (the American Standard Revised Version "drain") them out and drink them" (Psalms 75:8), i.e. God gives to the wicked the cup of wrathful judgment, which they must drink to the last drop.

See more on the meaning of Dregs in the Bible:
Dregs {Easton's Bible Dictionary}

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