No, standing from prone doesn't trigger booming blade's damage.
I don't see any specific rules definitions for "movement." Based on definitions in other editions, my opinion is that a character "moves" when they walk or use some other "special type of movement" such as jumping, flight, swimming, etc: when they move from their original position, rather than just standing up, dismounting, or otherwise reconfiguring their posture but remaining in the same place.
The spell doesn't say "if the target willingly takes any action." Therefore, attacks and environment interactions are probably fine. The spell doesn't force you to remain perfectly stationary like a statue. Given this, it seems clear to me that doing things that use movement but don't change your tactical position don't count.
Some places in PHB190-191 use the phrase "deduct... from your speed" instead of "spending [X] feet of movement." This might be a clearer way of looking at it. Some actions, such as standing up, reduce the amount you can move in the turn even though they are not "movement" for the purposes of triggering spell behavior.
For a useful corollary, it makes sense that mounted movement would still trigger the effect, even though you're not using your own movement or deducting from your speed.
Although it doesn't affect the lack of definition in 5e's rules as written, D&D 4e says "Whenever a creature, an object, or an effect leaves a square to enter another, it is moving." 4e rules don't apply to 5e, but they provide insight into the background of D&D rule design.