It has no specific meaning. It was arrived at through moving around syllables, seemingly at random.
(Allow me a personal eruption of triumph for a moment after hours of searching: I found it! I found it! I found it!).
JK Rowling revealed the story behind the creation of the word 'horcrux' in a diary entry on her old website dated September 29th 2006.
Sitting at my desk trying to invent a word yesterday brought back memories of the last time I did so. I had tried for days and days to hit upon the right name for ‘the receptacle in which a Dark wizard has hidden a fragment of his soul for the purposes of attaining immortality.’ Finally, after much transposition of syllables, I scribbled ‘Horcrux’ on a piece of paper and knew it was The One. But what if somebody had already used it? With some trepidation I typed ‘Horcrux’ into Google and, to my delight, saw what I was looking for: ‘Your search – “Horcrux” – did not match any documents.’
So anyway, yesterday I Googled ‘Horcrux’ again. 401,000 results. As you might imagine, this gave me something of a lift as I went back to scribbling nonsense words on the back of a takeaway menu.
It's understandable to expect there to be a hidden meaning to this word since Rowling imbues so many of her names and titles with hidden meaning, often originating in Latin or Greek words. Nevertheless, some of her creations were just, as Rowling herself puts it, "nonsense words" - and this seems to be one of them.
According to Rowling, 'horcrux' was just one of a long list of words that were under consideration. She seemed to come up with it through a process of trial and error, rather than purposefully creating a word from specific linguistic roots. She tried all sorts of words and went for the one that 'felt right'. In other words, she wanted a word that pleased her rather than a word that was etymologically pleasing.
She also says that she arrived at 'horcrux' "after much transposition of syllables". This underscores the previous point. She was moving syllables about, seemingly at random, until she stumbled across a word that she liked. She doesn't give any further detail about how she came to choose the syllables she did ('hor' and 'crux'). Presumably, she was trying a long list of different syllables out as they came to mind - not just choosing between 'horcrux' and 'cruxhor'.
Based on this account by Rowling, there's no link to the Cruciatus Curse. 'Crux' was just one of many syllables Rowling could have used.
Of course, the lack of an etymological meaning behind the word hasn't stopped people creating their own. Of the various ones that have been put forwards the one I like the most is this one that suggests that 'horcrux' is a hodgepodge of French and Latin, meaning 'the outside essence'.
The etymology of the word seems to be this: a combination of “hors” from the French “dehors” meaning outside and “crux” meaning “essence.” Thus, a Horcrux is a device for keeping your soul (the essence) outside your body.
Nevertheless, these constructions seem to be fan-based and created after the fact rather than anything that weighed upon Rowling's thinking at the time.