1. An inexplicable misfeature; gratuitous inelegance.
2. A hack or crock that depends on a complex combination of coincidences (or, possibly, the combination upon which the crock depends for its accidental failure to malfunction). “This hack can output characters 40--57 by putting the character in the four-bit accumulator field of an XCT and then extracting six bits — the low 2 bits of the XCT opcode are the right thing.” “What randomness!”
3. Of people, synonymous with flakiness. The connotation is that the person so described is behaving weirdly, incompetently, or inappropriately for reasons which are (a) too tiresome to bother inquiring into, (b) are probably as inscrutable as quantum phenomena anyway, and (c) are likely to pass with time. “Maybe he has a real complaint, or maybe it's just randomness. See if he calls back.”
Despite the negative connotations of most jargon uses of this term have, it is worth noting that randomness can actually be a valuable resource, very useful for applications in cryptography and elsewhere. Computers are so thoroughly deterministic that they have a hard time generating high-quality randomness, so hackers have sometimes felt the need to built special-purpose contraptions for this purpose alone. One well-known website offers random bits generated by radioactive decay. Another derives random bits from chaotic systems in analog electronics. Originally, the latter site got its random bits by doing photometry on lava lamps. Hackers invariably found this hilarious. If you have to ask why, you'll never get it.)