You can build a simple passive switch box easy.
http://www.digikey.com/product-detail/e ... ND/2054944Just get one of those switches for every input you want. Note, the link is for a 6PDT switch (the picture is generic). The 8-pole version of that switch is in stock at Mouser, but you only need the 6. One pin of the Framemeister's mini-DIN is unused, and you don't need to switch ground (just tie them all together across all inputs). Might as well save a little money and get the 6-pole version. Also, only worrying about 6 of the pins is less work.
The upside is that the switch box will be passive so you don't have to worry about powering it. The downside is that you'll lose out on nice touches like LED indicators for which input is active as well as the fact that you'll have to manually turn switches OFF (otherwise you'll have multiple signals outputting simultaneously, which is bad). But it's a dead simple build. Just connect each input to a switch, then connect all the switches in parallel to the output. Done. And since it's less components, you can afford to use a smaller box.
edit: Just thinking, if for some reason you'll also need a switcher for composite and/or S-Video, if you got the 10-pole version of that switch you can use the extra 4 poles to add composite and/or S-Video. Again, you can just tie all the grounds together so 4 pins would be enough for this. So each switch would control a 8-pin mini-DIN input as well as a composite or S-Video input.
Second edit: You should be able to easily see visually which switch is engaged, but if you really wanted indicator lights it wouldn't be too much extra work. You'd have to get the 8 or 10-pole version of the switch. Use one of the extra poles to turn an LED on/off. You can power the LEDs with two AA batteries. The batteries wont take up much room in the case, and you still wouldn't have to worry about using a power adapter (unless you don't care about that). If you choose the right LEDs, you could probably get a couple hundred hours of battery life. You could add a battery compartment without much trouble.
Third edit: This GIF shows the simple circuit for the LED (the side with the switch is the positive side of the battery).
The bus switch IC is pretty neat, but I wouldn't go that route unless I made a PCB for it. Using that particular IC in the link, you would need one per input. I don't know how many inputs you need, but I wouldn't wanna wire all that up by hand. You could get a PCB made that'll fit perfectly and mount inside a project box, so you wouldn't have to do the floating wires and hot glue mess like they did in that link. And it wouldn't cost much. OSH Park is $5 per square inch. The catch is that you have to buy a minimum of 3 boards, but if you're building multiple switch boxes this wont be a problem. Even if you're only building one, you should be able to off load the extra PCBs in the BST fairly easily.
http://www.digikey.com/product-detail/e ... ND/1154207Another edit: If using the CBT3244AD, I still see a problem with having multiple inputs activated at the same time. It would be neat to use push button momentary switches with a little extra circuitry to control the chip selection (so when you turn one IC on it turns all the others off). I wouldn't know how to do that, but this looks like the right track:
http://www.instructables.com/id/Use-a-M ... /?ALLSTEPSOr, if you don't mind the aesthetic, you could select each IC with a rotary switch. That might be kinda neat if you're going for a retro look.
Also, those ICs are 8 in/out, and since you would only need 6 for the Framemeister mini-DIN, you could use the IC to turn the LED on/off as well.
