shmoneh

The Hebrew word for the number eight is שמונה - shmoneh. A word that might seem to have the same root is shemen שמן - "fat, oil". Are they connected?

Strong seems to think so:

apparently from shamen through the idea of plumpness; a cardinal number, eight (as if a surplus above the "perfect" seven);

I suppose the idea here is similar to the slang phrase in English - "everything else is gravy".

And there are no shortage of drashot about Chanuka that connect the number eight (the days of
Chanuka) with shemen (the oil used in the miracle.)

But I'm inclined here to accept Horowitz's opinion. He writes:

Two different sounds are represented by the Hebrew letter ש (shin). One is originally and really ש (shin); the other is a "th" sound that coalesced into ש (shin) in Hebrew. Scholars write this second sound ת (tav). Here, if you have studied a little Talmud, you will be familiar with many Hebrew words that Have ש (shin) in them but in Aramaic are written with a ת (tav) ... So, therefore, don't try to connect the following roots that have ש (shin) in them ...

שמן (shamen) - fat
שמונה (shmonah) - eight. The ש here was originally a "th". In Aramaic the word eight is תמני.

By the way, according to Klein, the Hebrew word for octopus, tmanun תמנון - derives from the Aramaic tmanya תמניא (eight) and nun נון (fish).