If it requires any movement at all, then a surprised character can't voluntarily drop prone. Ask your DM if they rule that the movement or positioning of falling prone has to require appreciable effort or if it has to require any movement at all.
That a surprised character can't voluntarily drop prone, is covered in the following four paragraphs. I then immediately go into explanations on why each step is relevant to the terminology. After that, I propose a different reading of the rules that goes into the convoluted argument that you can drop your weapon without making a free object interaction which has resonance in this thread and likewise that you could argue that falling prone is no movement at all because there is no appreciable effort required to fall prone. At last, I raise the argument that even if you consider falling prone to take no movement at all you may not have the awareness to fall prone if you are surprised (this introduction is inspired by linksassin). Now to the argument:
Speed only determines how far you can move.
Speed, PHB 181:
Every character and monster has a speed, which is the distance in feet
that the character or monster can walk in 1 round. This number assumes
short bursts of energetic movement in the midst of a life threatening
situation. The following rules determine how far a character or
monster can move in a minute, an hour, or a day.
But the condition Surprise is explicit in that you can't move, at all, logically you can not do any of the Movement and Positioning under which Being Prone is listed (PHB 190).
There is a distinction in terminology that is made regarding Conditions, PHB 290-292.
Petrification lists the distinct states, Petrified, PHB 291:
The creature is incapacitated (see the condition), can’t move or
speak, and is unaware of its surroundings.
- Can’t move.
- Can't speak.
- Is Unaware (of surroundings).
Being grappled further indicates that a bonus to a creatures speed would resolve the issue of having 0 speed, it could, therefore, move.
Grappled, PHB 290:
A grappled creature’s speed becomes 0, and it can’t benefit from any
bonus to its speed.
Having 0 speed and not being able to move is not the same. The different states of being petrified showcase the difference in the agency that is left to you.
Is there a case for you being able to drop prone without being able to move?
Yes, there is!
There are two philosophies, according to this one you can drop your weapon and not count that weapon as an object and therefore make no object interaction because
letting go of something requires no appreciable effort. But picking it
As argued in this thread, the quote refers to this Tweet by Jeremy Crawford.
(The other is that weapons are objects)
So if letting go of an object costs you no object interaction because there is no appreciable effort required, then you could argue that there also is no appreciable effort required from moving from standing into dropping down - merely relax your muscles.
This perspective follows the Condition of being Unconscious, PHB 292:
The creature drops whatever it’s holding and falls prone.
Here dropping whatever (including weapons - whether they are objects or not)) is treated in the same manner as falling prone, indicating that there is some effort required in standing up.
Even if you do not count falling prone as a movement being Surprised also prevents reactions. Reactions such as the Shield spell, which happen so quickly that you can decide to cast Shield between being hit by an attack and taking damage from that attack. (Thank you AgentPaper for pointing this out explicitly)
What Surprise does not do, is: make you unaware (of surroundings) - compare to Petrified above. Therefore strictly reading: if you buy into the "no appreciable effort required" argument, then you would be able to voluntarily fall prone under the condition Surprised.