I'm looking to get a PC for my niece this Xmas. She's not terribly responsible and her house isn't kept clean (cats and dust will be an issue), but she's a good kid who needs a machine for school work... and, let's face it, Minecraft.

Any advice on a fairly indestructible PC tower to get her?

Anything cheap from Dell or Best Buy would be fine.

A chromebook would be a decent option, but getting Minecraft on it requires some work.

Something like an Intel Nuc sounds perfect (one of the newer ones).

I wouldn't get a laptop as they have multiple points of failure and you definitely need to spend above entry level to guarantee Minecraft will work well, a lot of cheap laptops have crappy Celerons with very poor graphics. If you go laptop, you will want a modern i3 minimum to make sure Minecraft will work well.

Most basic modern PCs will run Minecraft so that shouldn't be too much of a problem.

Might be worth getting a copy of Microsoft Office if they need it for school work. IDK if Schools still use Office actually but it's something to think about if that's the purpose.

Buying a lesser PC just for school work has never worked out for me, laptops/other PCs, I always wish I had more power to do more stuff with them once I get them.

Her age doesn't matter, to not only game but enjoy gaming she needs something solid like an i5 minimum in my opinion + SSD for the OS drive and an ok gfx card depending on what games she likes. Sure, if she's only into Minecraft and less graphics intensive games the GPU doesn't have to be a big expense in this case. But whichever you do go for, seriously get a PC with an SSD, it helps everything.

If you want a Free Office/Word-like software, I recommend LibreOffice. Its compatible with Word-files.

This just come from a place of knowing 11 year old nieces - If it doesn't have to be a desktop you might consider a laptop. She will want an Apple, again I know 11 year old girls, but see about getting her a nice looking laptop and a Apple sticker...if she has a sense of humor. Sure a desktop you build will be better in every way, but the mobility of teh laptop even just around the house might be nice.

I think a good deal now, that won't break the bank is Dell Inspiron 15 3000. It is about $380 but has a notebook i3U, 4GB of RAM, and 1 TB drive that is slow but generously sized. The Intel HD Graphics 5500 is enough for Minecraft on the 15" screen.

NOTE:
But look around for models you can definitely fine ones with 1920 x 1080 pixels and better HDDs, or upgradeable as other have said. The nice thing about a cheap laptop is it only needs to last two year,s and then they can get another cheap one that is updated.

A cheap Windows tablet or laptop and a gamepad for Minecraft sounds like the way to go. I can't really be more specific, since I've not kept up with the low-tier stuff recently.

A Raspberry Pi can run Minecraft.

It's cheap enough that it can be broken without too much worry, perfectly capable of being used as a basic desktop PC and has the added bonus of hitting the requisite 'edutainment' points with an 11 year old's parents.

A cheap USB keyboard and mouse (I use a Logitech K400 which takes care of both) plus a HDMI TV is all you need for the latest 3b Pi - It's wi-fi and bluetooth enabled.

I bought one of those for my mother last year. It's solid for sure, but that hard drive is painfully slow, even for a mechanical drive. Fortunately it's super easy to replace, so just make sure you get an SSD as well if you go this route.

I'd say an Alienware Alpha. But, I know 10 year old me pirated tons of crap and I killed things pretty often trying to free up 4k more EMS memory so I could use a joystick in TIE Fighter. So, if you go the PC route, make sure you make a backup of that hard drive for recovery...

If I wasn't big into gaming and 11 years old, I'd probably want a MacBook. Woot sells refurbished ones pretty often, and with HD 4000 integrated graphics, I'm sure it'd run Minecraft at 1440x900.

Woot's also been pushing older Shuttle PCs. I've seen em pop up a few times and have been tempted to nab one. There was an i3 with 8 gigs of ram and a 2tb HD for $199. Toss in a 1050 for $120 and you have a pretty beefy machine for a 11 year old. Only problem is they just popped up, so it might be a bit before they show up again. If they show up again.

We have a few Surface 3's on campus. They're actually pretty nice and they're fanless, so no need to worry about cat hair. The Surface Pro 4 with the M processor is also fanless and doesn't use slow emmc - probably won't notice it in daily tasks, but the Surface 3's slow drive can be felt at times. They both can run Minecraft. I don't know how I'd feel about a 11 year old having it, but my 37 year old ass carries his Surface Pro 3 every day in a backpack. If she were high school aged? I'd for sure say Surface. Taking notes in One Note and having the ability to record audio while your written notes put in time stamps is awesome.

Is her school using Office 365? If so, she should have access to Office 365 online and might also have X installations to other devices.