Meaning of "top" in "to sleep as sound as a top"

It means they slept quite soundly (that is, quite deeply). Note, an entry in Willis's Current Notes, on page 48 of the June, 1857 issue, makes the following claim about “sleeps like a Top” or “as sound as a Top”:

The saying is derived from the Italian in which language, the word topo signifies a mouse; it is the generic name, and applied indiscriminantly to the common mouse, the field mouse, or the dormouse, hence the Italian proverb – Ei dorme come un topo, or in English – He sleeps like a Top!

Edit: Another source (Varieties of Literature: Being, Principally, Selections from the Portfolio of the Late John Brady, Esq., 1826) on page 14, quoting from Gentlemen's Magazine, 1793, gives a similarly-worded etymology for the phrase, suggesting it might be the source for the 1857 entry. Gentlemen's Magazine also commented:

...we generally imagine the simile taken from the momentary pause of a peg-top, or humming-top, when its rotatory motion is at the height. But no such thing: the word top is Italian.

Tops were said to be sleeping as early as the late 17th century (and probably earlier) when they were spinning but not otherwise moving. This terminology is now uncommon for tops but still used for yoyos.

The usual form of this expression is "to sleep like a top". It means to sleep soundly, and is simply a simile like

he lies like a rug,

which uses two quite different senses of a word. Such similes should probably be considered a form of wordplay.

Here are some early citations of "sleeping top".

Charlotte Smith, in 1795:

... that can stoop
To follow, sportively, the rolling hoop;
To watch the sleeping top with gay delight,
Or mark with raptured gaze the sailing kite.

John Ayrton Paris, 1831:

if indeed the centre of gravity happens to be situated perpendicularly over the point of rotation, the top will continue quite steady, or sleeping, as it is termed, till nearly the whole of its velocity is expended.