I remember once hearing a liberal Conservative rabbi speak from his pulpit and challenge the congregation. He liked his sermons to be interactive, so he asked the gathered crowd whether anyone could explain, without resorting to the Documentary Hypothesis, why the Torah alternates between using the words "sack" (sack) and "amtachas" (pack) for essentially the same word in the Joseph story. Of course, no one had an answer and all were left with a convincing sermon about two textual sources for the story (presumably, although not stated, J and E).
This point is actually addressed within the Jewish exegetical tradition, although not necessarilly fully. Rashi (Gen. 42:27) merely states that the words are synonyms but later commentators explain that they mean different things. The following are some explanations, although they do not address in detail why the Torah sometimes chooses one word and not the other. I perform the exercise for no. 4 but the other explanations also need similar work.
1. The Ramban (Gen. 42:27) infers from/expands upon Onkelos that a sack is a large bag with many smaller bags inside it. The general bundle, i.e. all of the bags together, is called a pack.
2. The Ramban then offers his own interpretation, that a pack is a subset of sack. It is a large bag with openings on two sides.
3. The Abarbanel (Gen., p. 405) suggests that sack is a general term and pack is a specific term used for the same object when it carries food.
4. Abarbanel also states that there are those who explain that sack means a large bag in which they put large amounts of food and pack is a small bag that contained other items. The Malbim (Ha-Torah Ve-Ha-Mitzvah 42:27) also adopts this view, as does Nahum Sarna (JPS Torah Commentary 42:27).
According to this explanation, the sacks were filled with a lot of things, including a pack that had personal effects and, unbeknownst to the brothers, the money with which they had paid for the food. Yosef commanded his servants to put the brothers' money in their sacks (42:25) because he didn't really care where the servants put the money. They chose to put it in the logical place, the bags of personals, so the money did not get lost within the food. One of the brothers then opened his sack in order to take out food for the animals and, for whatever reason, also looked in his pack and found the money there (42:27). When they all emptied out their sacks entirely, including their packs, they found that their money had also been returned (42:35). And, thus, repeatedly in ch. 43, the pack is where they find the money and, in ch. 44, the silver goblet. The only question is why in 44:1 Yosef tells his servants to fill the brothers' packs with food and money. Shouldn't the food only go in the sacks and not the packs? Unless he wanted the servants to put food in the packs also, so the silver goblet would be better hidden.
5. The Netziv (Ha'amek Davar 42:27) suggests that pack is an expensive type of bag that is carried inside a standard, inexpensive sack.
While more exegetical work is needed to explain the specific passages based on these varying explanations, it seems that one need not resort to the Documentary Hypothesis to understand the use of different terms. This is also the conclusion reached by some modern scholars.
Gordon Wenham, Word Biblical Commentary 42:27-28:
Sarna suggests the pack was carried inside the sack, but this appears unlikely in 44:1 [but see above, no. 4 -- GS]. Rather, it would appear that sack (cloth) was the broader term and "sack, pack" a more specific one [see above, nos. 1 & 3 -- GS]. Traditional source critics used the two terms to distinguish two sources, but as Westermann (3:112) points out, the presence of both terms here in a single verse makes this a dubious criterion for source analysis.