I still haven’t found a reasonable etymology for the phrase (although M-W says it was coined in 1959 without explaining how).
As to what a think tank is, try looking up the grand-daddy of think tanks, the Brookings Institution at http://www.brook.edu/ and click on the «Message» and «About Brookings» links in the Welcome section on the left, just below the fancy intro headings. I think most of them are similar in that they are operated either as a “service” by a university or they are operated much like a university–Brookings, for example, is financed by endowments, (and notice their «edu» domain).
To join? Probably the senior fellows are recruited from specialists in the field who have published papers that the think tank approves (political think tanks tend to be–surprise!–left or right leaning). The support staff is probably recruited from colleges or the general public. If you scroll to the bottom of the Brookings home page, you will find a section on employment.
(I’m pretty sure, however, they have rules against hiring people named Joe Bob or Bubba. < eg > )
Sorry for proliferating the multiple threads. I saw this one with no response for two days and responded before I found the thread with 12 responses.