Tuesday/Friday
I like this puzzle, but I like the two different perspectives just as much. There are two current theories on which day the discussion occurred, either Tuesday or Sunday.
Both sides realize that the result of:
When the day after tomorrow is yesterday
is today+3 days. Now the girl places herself 'in' that day by talking about today+3 as 'today', and this is where the points of view split. In the second part of the prattle the girl says:
the day before yesterday was tomorrow!
We all know that that means today-3. In the original answer (Sunday) the today in today-3 is the same day that the prattle occurs. This means that today+3 and today-3 must be equidistant from Sunday, so the prattle must have occurred on Sunday. While in the second opinion we are still in the 'today' reached by the first part of the conversation when we talk about the day before yesterday being tomorrow. So 'today'-3 becomes (today+3)-3. From this we have to find a day which is as far from Sunday as 3 days later than itself. The solution to that case is that the prattle occurred on Tuesday.
Both are interesting interpretations. Both work, I think, but I would lean towards the Sunday interpretation because the hypothetical 'today' was in quotes while the tomorrows and yesterdays were not which leads me to think they were not hypothetical tomorrows and yesterdays, but real.