In the work that I do with coaching student leaders, many of them are asked to give extemporaneous speeches for the purpose of training them to think and speak on their feet and in the moment; which is something all of us do everyday in some capacity. So, here are my thoughts on effectively preparing for and delivering an extemp. speech (and in the context of an environment where the person still does have at least a few minutes to prepare their thoughts...)
Remember, this is extemp. speaking, not impromptu. This means that you should have already put together mini-chunks of information that you think you will be able to utilize during your extemp. presentation. This is also how you should organize your thoughts as you are preparing right before delivery. Think in chunks and just piece enough chunks together to fill the time and that flow well together.
Extemp. presentations should be more conversational in delivery. You aren't expected to have every word down perfect or know where to really place emphasis. Therefore, more relaxed verbal and non-verbal methods are more acceptable.
Even though conversational, the extemp. speech should not be sloppy. Because you don't have the words memorized in order, if you naturally have any fillers or weak words that you normally use, they will show up to haunt you during an extemp. speech. Therefore, you must be even more diligent about getting rid of your fillers (um, uh, you know, like, definitely, etc.) and your weak words (maybe, sort of, a little bit, probably, etc.).
Utilize stories! They are great time fillers and your words will come more naturally.
Utilize facts/data/statistics, but keep them intriguing and off-the-wall to keep the creativity level high (which is sometimes dampered because of the short preparation time.)
Spend 40% of your time preparing the outline of your chunks and 60% of your prep time memorizing the flow of chunks (get this flow very, very, very clear in your mind - say it out loud, write it five times, read it over and over, etc.) and saying out loud what you have planned. But, don't worry about writing it out or saying it word for word.
Use note cards to stay on track while presenting if you can. But you really only need one card with your chunks outline on it.
This is not a complete list, but it gives you some things to work on...