BogoMips (from "bogus" and MIPS) is an unscientific measurement of CPU speed made by the Linux kernel when it boots, to calibrate an internal busy-loop. An oft-quoted definition of the term is "the number of million times per second a processor can do absolutely nothing."
BogoMips can be used to see whether it is in the proper range for the particular processor, its clock frequency, and the potentially present CPU cache. It is not usable for performance comparison between different CPUs.
Both Dhrystone and Whetstone are benchmarks which measure computing power. The former concentrates on string and integer operations and therefore is more representative of the ALU speed, while the latter primarily measures the FPU performance. Depending on your application one or the other is more representative of what you're doing.