My search for information pertaining to the history of this printed piece turned up no results. Not to say there isn't any, I just couldn't find anything. I honestly never gave it much thought before and assumed they simply chose an aesthetically pleasing arrangement of the clock hands.
I found that this part first appeared in 1983 on set# 7824 "Train Station". This set and the next two released with this part (sets 6392, 4554) all show the clocks displayed as the time you mentioned, 2:57. This would seem to suggest that this was the time originally intended, as you had speculated. These sets, released between 1983 to 1991, had similar "travel" themes which made me question if 3:00 was a common time for trains and planes. The first set, #7824, DOES have 3:00 listed on it's schedule sticker. But, the next two do not.
Then, interestingly enough, in 1996 they were consistently changed for the next 6 sets to 11:43-ish, until 2004. Then, in 2007, they seem to become somewhat random in how they're oriented in the sets. I made a picture compilation of the first 16 sets to include this piece, in order of date released, with the orientation of the clock shown in the lower right:
I also was curious about other clocks that Lego has released in sets and if there was any sort of consistency with the times. A quick search on Bricklink for "clock" under "parts" brought up a very long list of pieces depicting a broad range of times:
It also appears they added an "updated" version of part# 4150px1 in 2009 with part# 14769pb001 having a different back on the tile.
Lastly, I totally agree with the speculation by Rin Rio-Oki of it being the "magical time" of being released from your tormentors aka "school". Regardless of whether any "facts" were to surface relating to the true origins of this piece, this theoretical meaning gives it character. Being a big Calvin & Hobbes fan since the 80's, I remembered there being a few comics that depicted the time Calvin got out of school. I eventually found a couple of the comics, and it turns out it was 2:30 instead of 3:00. My dream of naming this Lego piece the "Calvin Clock" quickly faded.
All rights to these comics belong to Bill Watterson. It is my hope that by using them here for a smile, and not for profit, that no infringement has been made.