Out of curiosity I tried plugging in a HDMI cable (£2.29 from e-bay with free postage) into an out-of-the-box BBB (I didn't upgrade the image first) and it seemed to work fine with a several-year old TV. Audio didn't work, but I understand that is a problem with the image.
The BBB has a USB port and HDMI, so the thought may occur by some to be usable as a media PC or general purpose computer for the home. I wouldn't advise that (probably better to buy something dedicated, like Apple TV - I have not tried that, or any low-cost Intel based PC for web browsing or media use, or for media PC possibly RPI since it supports full HD today - I've no idea of the performance of it, I've not tried it for such a use-case).
Also, as expected, the performance of the BBB, while high, is not a suitable replacement for a general web browsing device, even for young children. It could be useful for kiosk apps or learning or teaching/training with the HDMI and kb/mouse in classes but I'm only guessing.
I wouldn't advise HDMI and kb/mouse for development either; the fastest way remains to develop on a PC (running Windows or Linux for example) and transfer files across via SFTP for compilation and execution on the BBB via SSH. Most app development does not require export of the display, but if something was needed, X or VNC could be used (or an open source version of Remote Desktop - which seems to run faster than VNC for me). The built-in text graphical text editor seemed to work fine, and speed was reasonable for these types of apps and general window manipulation for the short period of time I tried it.
What could it be used for?
There could be low-cost commercial digital media player (e.g. signage) types of applications, kiosk etc as mentioned before, or the possibility to view a GUI before implementing on an LCD. The BBB resolution with the current image is 1280x1024, which as it stands would suit industrial LCD panels, but it scaled well with a sharp image on a normal television (see photos below) to suit signage applications too.
Again out of curiosity more than anything else, connecting a wireless logitech mouse to the board seemed to work fine (scroll, forward/back buttons worked too). I don't have a keyboard but apparently they share the same receiver so it is possible to have both the keyboard and mouse using the same USB port. I've no reason to assume it wouldn't work with the keyboard too. I found I didn't need any hub nor external 5V PSU to get the mouse and HDMI to work, but probably for a real use a 5V PSU to the BBB would be advisable.
Using mouse cut-and-paste I tried youtube. Some videos worked. There is clearly much frame drop but it is to be expected - Clearly Chrome or plugins are not using NEON: It would be more useful to try any media player software that was accelerated but I've not yet looked to see what exists.
GIMP was already installed as was this image in some folder: