[This documents a significant failure: to install a current version of Android, see Android 5.1-rc1 in VirtualBox]
I previously installed Android 4.3 in VirtualBox. Given that's a relatively old version, I went looking for a more recent x86 version of Android to install in VirtualBox: I found this blog entry and attempted to work through it. Apparently the author succeeded in the install, but I did not. (If you attempt this, ignore their advice to not install GRUB - I'm not sure how they managed that, as for me it produced a non-bootable virtual hard drive.)
My biggest issue with this is that Google doesn't provide a canonical site for the download of x86 Android packages (at least not that I've found). This would be a useful thing to both tinkerers like myself and developers. I believe they still provide developer images, but to use those you had to install an incredibly massive Eclipse-based package (~2GB last I checked) - it's not pretty. It wouldn't be hard for Google to provide a download site for x86 ISOs, and would provide some measure of security - as opposed to downloading a linked ISO from a random and untrusted site pointed out by some blog post you found, as here.
The ISO does boot, and does install - after a fashion. The trouble (for me) comes after you reboot the virtual box - I tried this on both Mac and Linux hosts, with similar results. When the system comes up, it puts up a window saying "Optimizing app 1 of 68" - there's nothing to interact with, you just have to wait it out. That process took about 75 minutes on both systems. But it doesn't get any better after that. You'll probably get to the language selection screen next, but regardless of whether you set VirtualBox's mouse to "USB Tablet" or "PS/2 Mouse" (the latter worked well with Android 4.3) you'll find the mouse incredibly tough to deal with and the system horribly laggy and unresponsive. If you leave it alone for a couple minutes, it'll reboot and start "Optimizing app 1 of 9" which will take between 30 and 60 minutes. Then you might get back to the setup screens, but miss it and it reboots and does more "optimizing." Mixed in to this were occasional errors "Unfortunately Settings has stopped" to which I just said "OK" and carried on. I managed to work through all the setup steps and get to an (still extremely laggy) Android desktop ... and after I left it alone it rebooted and started "optimizing" again. At which point I gave up.
I'm not clear what the problem is: I thought it might be that it wasn't detecting or using the network connection properly, but in the end I was fairly sure it was on the network. If you need an Android desktop in VirtualBox for running apps or any such, it seems to me you're still going to be stuck with version 4.3. If anyone has a better solution, I'd love to hear it.
After using a better but less private search engine (sorry DuckDuckGo, Google still offers better results ...) I located Android-x86 - Porting Android to x86 - which isn't a Google project. That explains the ISOs, and suggests that this site would be the best single source - wish they offered md5sums or any cryptographic hash on the ISOs. I'm downloading "android-x86-5.1-rc1.iso" and will report back later.