While there are probably some cultural reasons, are there any general reasons a 19x19 board may be preferred? Also, if, say, 21x21 was suddenly found to be in some way superior and a large number of player switched, would modern go strategy need to be significantly modified to account for the changes in gameplay?
A 5x5 board is interesting because beginners very quickly encounter all the basic rules of go - living, dying, territory, corners, sides.
A 9x9 board is interesting because the size is just right to develop fighting tactics.
13x13 lets you get into the mindset of occupying corners and linking tactics together.
17x17 might have been the next logical step, until people realized things were still too linear.. And then they brought it up to 19x19 ?
A fun fact is that early 19x19 used to have two black stones and two white stones, on the corner star points, making diagonals (d4 + q16, d16 + q4), which is nowadays considered to be an opening for fighting games. It may have been a first step towards a fully empty 19x19 board, or a more spiritual yin/yang reason, who knows.
19x19 board, people usually say, is interesting because it is the largest board size on which there are more points on the sides (under the third line) than in the center (above the fourth line). A 21x21 board has many more points in the center, and therefore the focus of the game would shift to fighting and living in the center, instead of trying to make territory on the sides -- since that would, obviously, just not be as beneficial anymore.
Even when you play on a 9x9 board, your basic strategy doesn't change much (or shouldn't); just the speed at which you effect your strategy.
I couldn't find concrete information, but the Wikipedia page mentions the fact the standard board had increased to 19x19 from 17x17 by the time it arrived in Japan and Korea. But it doesn't mention a reason for the increase in size.
My theory would be that the larger board size increased the time a quality game would take, particularly the beginning and middle phases.
It has to do with equi-distant handicap stone placement (in which each stone has equal 'effect' on the starting board position), and a balance between territory and influence aspects of the game (trying to start with each being equally important, as to make more complex fighting and strategies).
The 5 empty spaces from one star point to another and 3 empty spaces from each star point (excluding tengen) to the board's edge creates a balance in early Go in which the first four moves were played automatically (that comment by Trevoke was true by the way, the game didn't used to begin until stones were placed on each corner star-point). That said, I can find no information that implies the Japanese version of the game has ever used anything besides 19x19, and no information that implies (more recently than nearly two thousand years ago) that China has used anything different either, and even the oldest weiqi boards found are a mix of 19x19 and 17x17.
This is all moot though. At this point, it's no longer a matter of strategy or balance or what you think might be a better way of doing things- it's a part of culture and history. If 21x21 were suddenly found to be superior for some reason or another (though this is very hard to believe, as it throws off the balance of the game in a big way), a large number of people would not switch, so the premise is flawed. 19x19 is, and will continue to be, the most popular board size (as well as the official board size in every country in which its history is respected and professional players exist). It's not a matter of 'an aspect of the game', 19x19 IS the game. It's like asking if 3D chess will replace normal chess because of insert rationality here- it's simply not going to happen, because chess will always be chess.
If you multiply 19 x 19 you get 361, which is close to the number of days in a year.