What are your design requirements or goals? Any areas you expect to need to learn more about, or any areas you wish to focus on?
Because most of the Intel 8-bit microprocessors were so popular they are fairly easy to find as New Old Stock (NOS) or Pulled/Refurbished (reclaimed/recycled, usually tested). At least the 8080, 8088, 8086, and 8085 are available from distributors or obsolete / old-stock supplies.
Two others that are worth considering would be the Zilog - Z80 (technical description), and MOS Technology 6502 (technical description). The Z80 was created by Federico Faggin, who designed Intel's 8080, and made the Z80 a binary compatible (but enhanced) processor.
The Z80 requires fewer support chips than the i8080 with built-in DRAM refresher, a single (5V) voltage supply requirement (i8080 required +/-5V and +12V), and can be run at any clock speed up to its specified speed (e.g. 2.5 MHz for the first NMOS models).
Combined with 5V power supply, oscillator clock (e.g. TTL/CMOS oscillator "can"), Static RAM (rather than Dynamic RAM, DRAM) you only need a ROM/EPROM/EEPROM and interface (e.g. interface chips for parallel or serial interfaces) to built a minimal microcomputer.
One fairly popular book on building your own microcomputer using the Z80 was Build Your Own Z80 Computer (available with copyright permission of author/publisher) by Steve Ciarcia. Also check out the (dated) alt.comp.hardware.homebuilt FAQ, and the N8VEM community for additional resources and references.
The MOS 6502 was the another popular microprocessor, in part because it was so much cheaper ($25 USD) than the Motorola 6800 which was originally $300 USD circa 1975. It was used in popular systems such as the Apple I from Apple Computers, Commodore KIM-1, PET, and Vic-20, BBC Micro, and the Commodore 64 and Atari 2600 game system used 6502 derivatives (6510 and 6507). So there is a lot of material available from retro-computing and retro-gaming people online, and parts. The 6502, like the Z80 was produced by several sources (i.e. second sourcing) including Rockwell in additional to the primary designer / manufacturer, MOS Technology.
If you have a particular (strong) interest in the x86 or IBM PC / XT history then a 8088 or 8086 might be a educational target to consider. Otherwise I would lean towards the Z80 as my first pick, and the 6502 as my second choice due to parts availability and resource material availability.
The range of options is unlimited from a microcomputer built from almost exclusively discrete transistors, to a 25MHz 32-bit MC68030 workstation.
Tradition has a strong pull. But so does interoperability. Pretty much every existing file format and communications protocol operates on bytes. How do you handle these in your 7-bit microcontroller?
The PIC gets away with it by having the instruction space entirely seperate and programmed in advance from outside. There is some value in bit-shaving the instruction set, as it's the one thing you get to control yourself as a microprocessor designer.
If you want an extreme architecture, you could Huffman code the instruction set, giving you variable length bitness.
Best Answer
Think about it. What exactly do you envision a "256 bit" processor being? What makes the bit-ness of a processor in the first place?
I think if no further qualifications are made, the bit-ness of a processor refers to its ALU width. This is the width of the binary number that it can handle natively in a single operation. A "32 bit" processor can therefore operate directly on values up to 32 bits wide in single instructions. Your 256 bit processor would therefore contain a very large ALU capable of adding, subtracting, ORing, ANDing, etc, 256 bit numbers in single operations. Why do you want that? What problem makes the large and expensive ALU worth having and paying for, even for those cases where the processor is only counting 100 iterations of a loop and the like?
The point is, you have to pay for the wide ALU whether you then use it a lot or only a small fraction of its capabilities. To justify a 256 bit ALU, you'd have to find an important enough problem that can really benefit from manipulating 256 bit words in single instructions. While you can probably contrive a few examples, there aren't enough of such problems that make the manufacturers feel they will ever get a return on the significant investment required to produce such a chip. If it there are niche but important (well-funded) problems that can really benefit from a wide ALU, then we would see very expensive highly targeted processors for that application. Their price, however, would prevent wide usage outside the narrow application that it was designed for. For example, if 256 bits made certain cryptography applications possible for the military, specialized 256 bit processors costing 100s to 1000s of dollars each would probably emerge. You wouldn't put one of these in a toaster, a power supply, or even a car though.
I should also be clear that the wide ALU doesn't just make the ALU more expensive, but other parts of the chip too. A 256 bit wide ALU also means there have to be 256 bit wide data paths. That alone would take a lot of silicon area. That data has to come from somewhere and go somewhere, so there would need to be registers, cache, other memory, etc, for the wide ALU to be used effectively.
Another point is that you can do any width arithmetic on any width processor. You can add a 32 bit memory word into another 32 bit memory word on a PIC 18 in 8 instructions, whereas you could do it on the same architecture scaled to 32 bits in only 2 instructions. The point is that a narrow ALU doesn't keep you from performing wide computations, only that the wide computations will take longer. It is therefore a question of speed, not capability. If you look at the spectrum of applications that need to use particular width numbers, you will see very very few require 256 bit words. The expense of accelerating just those few applications with hardware that won't help the others just isn't worth it and doesn't make a good investment for product development.