In my opinion, "once" must mean "one time" here. "Once" can be used as a conjunction, meaning "as soon as" or "when", but when it is used like that it comes at the beginning of the clause. There is already a conjunction here, "When", and "once" in that sense would be redundant, as well as in the wrong place. In this position, following the conjunction "when", it must mean "one time".

In effect, the meaning is virtually the same: as soon as Satan managed to make Eve distrust God's word, she yielded; distrusting His word just once, the first time, was enough to make her yield. But the sense of "once" is, must be, the latter.

"Once" is being used here in essentially the same way as in this famous quotation from Abraham Lincoln:

"If you once forfeit the confidence of your fellow citizens, you can never regain their respect and esteem"


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Note added at 3 hrs (2012-05-02 11:52:44 GMT)
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Sorry: the word "to" is missing in my answer: it should read "had brought her to distrust (God's word) one time".

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Note added at 3 hrs (2012-05-02 11:53:25 GMT)
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I agree with your addition of "just": "just one time". Just once was enough.

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Note added at 3 hrs (2012-05-02 12:04:47 GMT)
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"Once" can be used in the sense of "even for a moment", and this may be implied here:

"But if you once doubt or hesitate, nothing will happen. This is illustrated by the story of Peter walking on the water. He doubted for a moment and just then he began to sink (Mt 14:28-31)"
Albert Nolan, Jesus before Christianity, p. 39
http://books.google.es/books?id=2UWm30LdqEYC&pg=PA39&lpg=PA3...

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Note added at 7 hrs (2012-05-02 15:41:05 GMT)
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"When he" is understood before "had brought her once to distrust":

When he had shaken that, and [when he] had brought her once to distrust: then she yielded."

But if you sustitute "when" for "once", it's nonsense:

"When he had shaken that, and had brought her when to distrust: then she yielded."

I repeat, "once" in this position cannot mean "when", and "when" is already present.

The position of "once", before the infinitive, is characteristic of older texts like this, and formal modern usage. Normal modern usage would be to put it after the infinitive:

"When he had shaken that, and had brought her to distrust [just] once: then she yielded."