The 4004, the very first microprocessor kind of just happened. The designers started out to design a chip set for programmable calculator rather than a processor. They designed a general purpose processor instead. That led to several other improved microprocessors including a 16 bit one the 8086. Then in a kind of step back Intel produced a microprocessor a year later, on July 1979, 8088. This was internally the same as the 8086, a complete 16 bit processor, yet external interfaces were all 8 bits. Looking back that might look like a retrograde step. IBM went ahead and used the processor in their IBM-PC product and the rest, as they say, is history!
If you looked at the way a slew of processor coming out of the major manufacturers and the way support chips catch up, it would be hard to believe it used to take time for the support and peripheral chips to catch up. One of the decisions PC manufacturer wanted to implement with the PC, was to use existing technology as far as possible. They went ahead and used the 8088 for that reason. With the existing support chips in 8 bit yet the internal architecture of 16 bit gives you benefits to an extent. At least it would be better than using a processor that is 8 bits inside out. The 8088 had a 8 bit data bus instead of the 16 bit bus of the 8086. Internal 16 bits registers and the 1 MB addressing capability was retained.
The 8088 CPU took two read or write cycles to read or write 16 bit data instead of one as in 8086. But these bus changes made it compatible with 8080/8085 support chips. The 8085 related chips were available in a wider selection, they were stable and also priced much lower than the newer 16 bit support/peripheral chips.
One would think it was logical to use a 16 bit processor for the PC. That actually came later. Processors in 16 bit were already available. The 8086 was around for a year already at the time 8088 was announced. The PC came about couple of years after the coming out of 8088. Motorola 68000 was on the scene too. Many considered the M68000 processor a very well designed 16 bit processor, designed from ground up. Existing, stable technology was the key decision about the PC and they stuck to it rather than go with newer technology.
At this time the processors still did not accommodate the complexity of floating point math operations in hardware. 8086 introduced a numeric c-processor, the 8087 that works with 8086 to provide floating point operations to the 8086. Interestingly the 8088 too could work with the 8087 and provide the math support. Thus there was no restriction that the PC could support office productivity applications only. As it happened later, scientific applications like the AutoCAD too would run on the PC. The PC was ready for the applications revolution on either genre and look what happened!