I recently re-read Nicholas Kaldor's article "The New Monetarism" from 1970. This contained an early statement of the principle that money should be seen as endogenous, in response to Friedman and his followers, who appealed to the idea of an exogenous money supply. Kaldor writes "The explanation ... for all the empirical findings on the 'stable money function' is that the 'money supply' is 'endogenous' not 'exogenous'."
Two points strike me from this paper:
The first point is that, for Kaldor, the question over the exogeneity or endogeneity of money is all about the causal relationship between money and nominal GDP. The new monetarists that were the subject of the article (we'd probably call them old monetarists now) argued that there was a strong causal direction from changes in the money supply to changes in nominal GDP, with the latter manifesting itself purely as changes in the price level in the long run.
Endogenous money in this context is a rejection of that causal direction. Money being endogenous means that it is changes in nominal GDP that cause changes in money or, alternatively, that changes in both are caused by some other factor. This is interesting because nowadays it seems to be quite common to use the term endogenous money to simply talk about the idea that 'loans create deposits', even in the context of models where the deposits so created have a strong casual link to nominal GDP. This appears to me to be almost the opposite of what endogenous money was originally about.
Secondly, Kaldor's analysis is based on seeing money for its function and what it does, rather than identifying a money supply with a particular asset class. As long as policy works to accommodate the demand for money, we might expect to see a perpetuation in the use of a particular medium - bank deposits, say - as the primary way of conducting exchange. But we would be wrong to conclude that bank deposits and money are one and the same. That they appear the same is only because it is convenient for them to function that way and because it has been allowed to happen. But any stress on that relationship will simply mean that bank deposits will no longer function as money in the same way. The practice of settling accounts will adapt, so that we may need to revise our view of what money is. Money cannot be captured in the concept of a "money supply".
UPDATE: Since posting this I noticed that John Cochrane has just done a post on Greece, including a comment on the use there of the rolling of post-dated cheques to deal with business to business payments. Cochrane writes "Money is created when needed, apparently." I'm not sure whether the apparently is intended to be ironic.