TL;DR - Just about any square taper bottom bracket that has a 68mm shell and roughly 116mm spindle should work fine. You should be able to find one for very little money at either your LBS or online.
As far as I can tell, the derailleur mounting to the bottom bracket doesn't make a lick of difference in selecting your bottom bracket. I've never actually used one, but from what Sheldon Brown says and the images I've found, it seems like it's just a ring that is held in place between the bottom bracket and its shell.
There are four things to worry about when replacing your bottom bracket:
- The threading
- The shell width
- The spindle type
- The spindle length.
Your bottom bracket appears to have standard English threading. This is by far the most common type of threading on the market. Just make sure that they one you buy has English threading (or, at least, doesn't list some other type of threading) and you'll be good on that front.
The shell width is the "68" in that 68x116 measurement that you mentioned. 68 and 73mm are the most common shell sizes. Many bottom brackets are sold to fit either one, some are specific to one or the other. The universal ones have a spacer that you just leave out for a 68mm shell.
Your bottom bracket appears to be a square taper bottom bracket. This means that the spindle is square and tapered. Sheldon Brown has an excellent image comparing different bottom bracket spindle types. You're lucky in that square taper bottom brackets are both common and inexpensive. There are also varying lengths of taper. Yours appears to be a JIS square taper, which is again, incredibly common and inexpensive.
Finally, you need the spindle length to match (more or less). The spindle length is the "116" in your measurement. I say it needs to more or less match because often have some wiggle room on a geared bike. Whether or not you have wiggle room, and how much you have, will depend on a lot of other factors. If you can, it's easiest to get an exact match.
For the sake of completeness, I should mention that there are also external bottom brackets. But those are a whole different ball of wax designed to work with a completely different crankset and probably not worth worrying about in your present situation.