The Hebrew word for six is shesh שש (masc. shisha שישה). The Hebrew and English words sound similar, and in other languages there is even a greater resemblance - seis in Spanish, sheshi in Lithuanian, and shesh itself in Persian.
(Persian is the origin of the shesh in shesh-besh, the Hebrew term for backgammon. Besh is five in Turkish, and according to Stahl, those two numbers are used because they are the highest ones on the die, and are taken from two separate languages because of the rhyme.)
Is there a connection between the Semitic languages and the Indo-European ones here? There are those who use this to prove their theory that Hebrew is the source of all languages, but most find the general theory far-fetched, and it is included here in a list of "false cognates". This article , from the 1911 EncylcopediaBritannicaa claims that:
Six is in Hebrew shesh, almost exactly like the Sanskrit and modern Persian shash, the Latin sex, &c. But the Indo-European root is sweks, or perhaps even ksweks, whereas the Semitic root is shidth, so that the resemblance is a purely accidental one, produced by phonetic change.
However, in the sci.lang newsgroup there was a serious discussion about the development of shesh. One poster claimed there that Indo-European borrowed the word from Semitic. But haven't we said earlier that words for numbers should be so basic that no borrowing would be required? His response is that perhaps the original Indo-European counting system was base 5 (the number of fingers in the hand), similar to Roman numerals. I have no conclusion one way or another. Perhaps a reader has access to further research on the subject?
Two homonyms that aren't related to the number six are shesh (the fabric) and shayish שיש(marble). Shesh the fabric refers to white linen, and despite the drasha in Yoma 71b that connects it to the number six (a six ply linen thread) - Klein and Steinberg point out that both shesh (linen) and shayish are originally Egyptian words. (Perhaps related to each other, both being white?)
One interesting possible derivative of shesh is the Hebrew word for lily - שושנה / שושן shoshana (or shoshan). This is a common name for women in Hebrew, and is the source of the English name Susan. Ibn Ezra connects shoshana to shesh in his commentary on Shir HaShirim 2:2 -
It is a white flower of sweet but narcotic perfume, and it receives its name because the flower has, in every case, six [shesh] petals, within which are six long filaments.
Klein backs up this theory, tracing shoshan to the Akkadian shushu - six-sided.
From this picture (click for larger version) we can see exactly where the name came from.