I wonder if the word is: Vademedum= a useful thing (changed to person) to have on a trip.
I immediately thought "steVedore"-- a ship loader. Their expertise traditionally includes figuring out how to jam all the cargo in the hold (not to mention keeping the ship in balance while they do so...). There's a great description in Richard Henry Dana's "Two Years Before the Mast" of the process of cramming thousands of stiff California cow hides into the hold of a Boston-bound ship in the 1820s...
I thought the caller started out by saying the word just meant someone good at something, and only later did the discussion zero in on someone who was specifically good at packing.
My first thought was "maven", but that would probably have been prefaced with the skill the person excelled at. After a minute or so, I hit on "savant". Surely that describes people with an ability that even they themselves can't explain.
I immediately thought of the term horror vacui which in art represents a fear of empty spaces. I don't know if the father who commented on her packing skills suggested she was exhibiting a classic example of horror vacui—I've never heard it used in casual, non art related, ways so I admit it's a stretch. I wonder of vacui or vacuum is part of the actual term she heard.
I like "vacuum-packer."