Expanding slightly, there’s a permutation that needs to happen in order to efficiently perform a DFT in-place (and the same approach is often used even when the transform is out-of-place). For power of two sizes (one of the most common cases), that permutation is precisely the same as a bit reversal of the indices.
Besides the FFT addressing, I have encountered SPI devices that wanted to be addressed in LSb-first manner. Some micros let you choose which end goes out first, some don't. In the latter case you find yourself swapping bits in software.
Passwords for the RFB protocol (VNC) are used to DES-encrypt a challenge token sent by the server. However, each byte in the key is reversed before it is used for encryption.
Of course, this happens only once per connection so there's generally not much of a need for it to be particularly fast.
Endian-ness would be reversing bytes, not bits within bytes, like 0x1234 -> 0x3412. What we're talking about here would be more along the lines of: 0b0010001 -> 0b1000100
The most obvious application I can think of for reversing bits within a byte would be for image processing applications, such as mirroring an image horizontally, or making kaleidoscopes. There are probably signal processing applications, too...
It was at one time standard to store each bitplane separately. If you had a 4-color 320x200 image, for example, you'd have one page of VRAM that stored a 320x200 1-bit image, holding all the bit 0s, and another, exactly the same, holding all the bit 1s. And so on up to as many bit planes as required. (That was one usual arrangement, but there are other options - e.g., interleaved bitplanes and/or no video RAM as such.)
Separate bitplanes were very annoying in many respects, and the demise of the approach was probably regretted by only a few. But storing each bitplane separately does have one major advantage: you can just write all your algorithms to operate on 1-bit images, and they automatically run at any bit depth. Just run the routine once for each bitplane you're interested in processing.
(That may also mean the hardware is simpler to implement - one plausible excuse for its ubiquity. Not my field of expertise though...)