A colleague and I were reading a corporate memo that contained the phrase 'redouble our efforts'. His claim was that the word redouble was equivalent to double and simply nonstandard. (Similar to, say, using irregardless in place of regardless). I've heard this phrase fairly frequently, and don't see anything wrong with it in particular, but I was wondering if perhaps there was some style guide that prescribed against it or something similar that marked a preference for simply doubling our efforts rather than redoubling them.
"Redouble" is a valid word in its own right, not non-standard at all. Its use is uncommon, to be sure, seen primarily in the idiomatic expression that you reference. Technically, according to its etymology (french "re-" + "doubler"), it should mean either "to quadruple" or "to double a second time." However, it has over time acquired a broader meaning as "intensify" or "strengthen."
The word redouble comes from French re- plus doubler, "to double". So to redouble one's efforts really means to double them again, to further emphasise or reinforce them beyond whatever doubling you may have done before. However, I'd go so far as to say that in corporate jargon, and even in common usage, it really is just a poor substitute for double or increase.
The expression can both mean to double and to double again. See thefreedictionary.com/redouble.
From the situation it should be evident if it means to double something that was doubled before or simply to double something. If it's not evident, the expression should naturally be avoided.
I think "redoubling" has an illustrative quality in that you can visualize folding a cloth over twice, the thickness increasing by four. This of course is accompanies with the area of the cloth decreases by the same factor.
Consider when someone says "we must redouble our attention", they are directing the group to shift their attention from a broad area, to a much smaller area indicated by the group's leader. The result may not be as intended, since the previously "attended" areas were probably also important.
So imagine a table covered by a table cloth being "redoubled" (or folded over twice) and left to lay were an especially messy eater sits. Or, perhaps the host simply pulls out three more table cloths and spreads them onto the table.
Clearly, a redouble is a double doubling of one which means it is appropriate as a second doubling. Then, technically, it is a doubling but in proper context signifies a quadrupling and is not a redundancy which of course is understood to be an unnecessary doubling. Giving it its broadest meaning possible, it can refer to any integer power of two beyond one.