Bitcoins value is made up of four parts :
1/Mining is not free , it costs electricity and significant processing power to mine them, but the process of mining adminisiters the system, the equivalent of the worlds banks, and financial services, all transactional and record keeping.
2/ The value between fiat currencies and bitcoin is not that different. I can buy a growing number of items with bitcoin and have no trouble selling them.
3/ Gold, diamonds, gems etc are valuable, mainly because they are rare on the surface of the earth. Sure gold has some value in electronics, the value of bitcoin is closely tied to its finite amount of 21M units. Fiat currencies don't until significant banking regulations permit, have that value
4/ It is an alogrhythm, it can't be manipulated by politicians. The media influence on the population is however a factor , but thats a cognition problem not one with the currency :)