For example, the new B787-8 and B787-9. Is there any significance in why they started with -8, versus the older 737s which started from -100? Or B757-200/300?
Chinese superstition. The same reason Airbus went with the 380 number when 350 would have been next in line.
The Chinese sign for 8 looks like an up arrow (八) and 8 is the lucky number in Chinese. Similarly, 4 (四) is considered unlucky because the word for it (sì) sounds similar to the Mandarin word for death (sĭ).
Boeing and Airbus management court their prospective Chinese customers wherever they can because they expect their biggest sales in the next decade from that part of the world.
Originally, Boeing used three-digit numbers for their subtypes, and the last two digits represented the airline. Pan Am's number was 21, for example, so a Boeing 707-100 made to Pan Am specs was a 707-121.
The numbers are indeed chosen mostly for marketing. Although there is an element of logic and organization of models, as with other products, the name is primarily a marketing tool.
The 787-8 emphasizes the number 8. This number has positive connotations in some Asian cultures. Including the two most populous contries in the world, Asia represents a huge market that has been growing rapidly, and this means big opportunities for aircraft manufacturers. As discussed on another question, Airbus and Boeing are both looking to gain orders from this growing market. The launch customer for the 787-8 was a Japanese airline, ANA. Almost a quarter of 787 orders have come directly from airlines in Asia.
Starting with 8 also makes the type seem more established than starting back with -1 or -100, and allows the 747-8 to share the same number to connect the technology shared between the two, mostly in the engines and wings. The 747 connection to Asia is even more apparent than with the 787. Airlines in Asia represent a large portion of past 747 deliveries, which provides an opportunity for them to upgrade to the newest version. Although the launch customers for both cargo and passenger versions are European, Asian customers represent almost half of the orders. Using a single number also differentiates their models from the competition, but also seems to have confused some people that are used to the old convention.
Starting with the 707, Boeing planes started with the -100 and added variants from there. This includes the 727, 737, and 747 as well. The 757 started with the -200 model because this corresponded to the approximate capacity of the 727-200 model it was replacing, which is where they saw the most demand. The 767 started with -100, but this model was never offered, being too close to the 757, so the 767 also starts with -200. The 777 is a similar situation, where a -100 was considered but never offered. The 737 has now run out of numbers, and rather than go up to -10 like the 787 or -1000 like the A350, they chose to go with -7 through -9. This matches the scheme of the other newer models while also relating to the corresponding NG versions of -700 through -900.
The A380 is in a slightly different situation, skipping both type and model numbers to start at A380-800. Although by far the most orders are from one particular airline, customers in Asia still represent almost 20% of the orders. Asia would seem to represent a prime market for a large airplane, with both high traffic routes and long international routes, but like the 747-8 the A380 has not received as many orders as expected. While it's unlikely that either Airbus or Boeing will disclose details of their marketing strategies, it's clear that the number 8 is a positive number in many Asian countries, both manufacturers have chosen to emphasize the number on all of their newest models, and the region represents a very important market for these aircraft.