My party is very likely going to end up in prison soon. The problem is that the party contains a droid PC. In world, it really doesn't make sense that the droid would go to prison with them. They would most likely just wipe or destroy the droid or maybe shut it down and put it into storage as the party's property, but sending the droid to prison feels super weird. I can always just ask my players to suspend their disbelief, but I'd rather not. Anyone have thoughts on how I might sort that out?

Since you naturally want to keep the player involved, fit the droid with a restraining bolt, and have it working in janitorial and/or the prison mess hall because they’re short handed. Maybe the prison staff was ordered to wipe the droid’s memory first, but didn’t because the staff likes watching the droid and the other PCs suffer under the droid’s indignity. Or wipe him, but allow the players to flip a Destiny point to have a backup that can be installed.

I like Nytwyng's idea the best. Consider this another option or adding some details to what Nytwyng and you were already considering.

You mentioned the droid being property of one or more of the other PCs. As property of an imprisoned person, one option you mentioned is storage, which doesn't work at all for the player of the droid. Another option though would be confiscation by the government. I'm assuming Imperial here, but don't know your era. But, it could easily be justified as "civil forfeiture of assets" from the criminals. So, the droid would be packed up, memory wiped, then sent off to a government facility that needs it. Use the Destiny flip to cause an error in the memory wipe. It wasn't successful, but only the droid realizes this and can play along. Perhaps give the droid some Setback for a while regarding past memory recollections.

Then, have the droid put to work somehow nearby or even in the prison. I suppose this depends on the type of droid. An Astromech? She/he is working at the nearby landing platform. Combat droid of some sort? Security at the prison.

The other imprisoned PCs now have an "outside man" to help with their eventual escape. The droid PC has to worry about the memory wipe failure being discovered while contacting the other PCs somehow and formulating a break out.

Ep VI handled this pretty well. Nytwyng is on the money. Droids are dead useful and can easily be recommissioned. if it came down to "why didn't they memory wipe me?" just have the equipment conveniently be broken or put him on the "schedule" which adds a nice little timer element.

As a GM, the most appealing thing about wiping the droid’s memory would be filling in the droid player, but not the others, before that first session begins. Let them encounter the droid performing its new duties, and think, “Hey...we’ve got an ace in the hole here!” Then, make contact with the droid...who doesn’t recognize them. It adds another element of rescue to the story. But, I also wouldn’t want to totally eliminate any character development, so - if the players don’t think of asking for a memory backup with a Destiny point, I’d just say it exists. Unless, that is, the player wants to play with the blank slate.

Run a slicing based encounter for the droid character to avoid a memory wipe. It's possible to win and keep their data, but also possible to lose. In a loss event, give them the option of "running away" and load their personality to another droid that happens to be plugged in nearby. Turns out it's a mouse droid and now they have to follow their old body as it goes about its menial tasks with a blank memory.

You have a whole comedic setup of the other characters trying to interact with their droid friend while some annoying mouse droid keeps pestering them!

I did this a couple years ago.

The party was taken into the equivalent of county lockup, so certainly looser than hard time. But for what it's worth, I had the droid player's character fitted with a restraining bolt and taken to one of several garage-like droid pens not unlike the impound lot. His "jailer" was a slow, soft-spoken Kitonak used for maximum comic relief. Left in the dark with a roomful of deactivated droids, the droid carefully removed his bolt and searched the chamber until he found another droid worth switching back on. Before long the party members' allies had bailed them, and since the reactivated droid was years-over unclaimed, the player was able to pay to get him out as well.

For the droids memory not being wiped you can say the prison needs to download his memory for their case or to find other associates of the group. The person to do that will not be there for 3 day, so until then he will be kept in storage with the groups other personal belongings.

So the droid is evidence, locked up in the impound cage? I like it. Maybe some sort of lonely sentient appliance is locked in there with him that can help disable the restraining bolt? An automated hairdresser given a personality to gossip with it’s clients (and with appendages capable of picking the restraining bolt).