Why are the pips on dice organized the way they are?
This is a really interesting question and just spent last hour googling around for various thoughts.

The first thing is why pips and not numbers. This is because the invention of dice predates the invention of numbers. source

According to the wikipedia article on pips its notes that the pip designs are 'easily countable'  Why the particular patterns for the numbers I don't know the fact is if you roll a 5 you recognize the pattern straight away.  Its worth noting the the pip design on a six sided dice is the same pip pattern on standard playing cards for values Ace-6.  Someone may be able to explain the origin but I imagine its a similar answer to why is are Letters shaped they way they are?  I's assumes the pip pattern is just something that's evolved over time that we then recognise.   if a dice (or a playing card for that matter) just displayed pips in a long row then counting them each time would be very tricky so the fact they evolved into a pattern that we regonise without counting the pips is probably where it came from.

Finally, why do numbers on opposite sides add up to the same number?  According to to this article Dice were not always like that, apparently some Roman dice were arrange with opposite sides adding up to prime numbers.  ie 1 opposite 2, 3 opposite 4 and 5 opposite 6.  This article states that if a dice is random it really doesn't matter what sides the numbers are on,   However the article suggests the rise in opposite sides adding up to 7 appears to be that was because it is perceived to be more fair, even though its still random what number comes up.  From the article:-


  “We think users of dice also adopted new ideas about fairness, and chance or probability in games,” said Eerkens in a statement. The numbering style changed from the prime number configuration (1-2; 3-4; 5-6), one heavily influenced by popular ancient Egyptian numerical arrangements, back to the style “where opposite sides add up to seven (6-1; 5-2; 3-4).” Eerkens suggests that while the direct cause for this shift is unclear, it could possibly be related to an gradual effort to make fair and balanced dice.


Also I should say that someone else with more historical or archaeological knowledge than I might be able to give a different answer.  I'm really intrigued by the idea that opposite numbers on a dice adding to same number is perceived to to be more fair that any other arrangement despite it not making any difference.