You can think of stone here as a word that closes a door. Dead already says the thing; stone says there is no crack left open, nothing that is partly otherwise, nothing that might still come back. English uses stone this way in a small family of words — stone-cold works the same. So both meanings are really one move: take a flat, final word and push it all the way to the end.
可以想成 stone 在這裡是一個「把門關上」的字。dead 本身已經把話說完了，stone 則是說：沒有留縫、沒有一部分不是這樣、也沒有回來的可能。英文有一小群字都這樣用 stone，stone-cold 就是同一種造法。所以兩個意思其實是同一個動作：把一個本來就很硬的字，再推到最底。
From the word’s history
The supplied source records stone dead from Middle English stone-dede / stan-ded, equivalent to stone + dead, and compares stone-cold and stonehard — a small family in which stone works as an intensifier.
手上的資料把 stone dead 記為中古英語 stone-dede、stan-ded，等於 stone 加 dead，並且拿 stone-cold、stonehard 相比。也就是說，stone 在這一小群字裡的角色一直是加強語氣。
Wiktionary, "stone dead"