Q:

Asked to demonstrate I am efficient on job application

I have been asked to give examples in a job application of things that show I am efficient.  A friend suggested that I should only give examples of ways in which I have done things on time or earlier. 
I think it would be good to give examples of things I have implemented that have made a task shorter than it would otherwise be (e.g. noticing that certain parts of the task can be skipped). 
By way of examples, the first type of efficiency would be handing in a thesis early, where as the second type of efficiency would be automating various parts of writing a thesis (contents, index etc.). 
The job is for a think tank, and my background is that I am finishing a PhD in maths. 

A:

I'd say that either type is fine, and examples that cover both are even better. 
For example:  

After writing my first thesis and handing it in on time, I found that certain parts of it could be automated to save time.   In between thesis papers, I wrote an automation script.   On the next thesis paper, the script saved me 20 hours of work.   The script took 40 hours to write, so I earned the time back after 3 thesis papers (saved 60 hours at that point!),  making it a net win

I've seen that people fail to realize that "efficiency" is a double edged sword.   Profound, long lasting efficiency improvements often take an initial investment up front, that is LESS efficient than doing it the original way.   Showing that you have a sense for how to make your work more efficient, but that you remain sensitive to the idea that in the end, the bottom line is getting things done quickly with the same quality, is really important.