If you want to analyze large sets of data, you can also work on the data you probably have right now or may have easily.
Example 1: web site logs
If you have a website and you preserved all the logs for the last five years, chances are, you have a big amount of data. Being a victim of DOS/DDOS helps a lot too.
This sort of data is also funny to play with:
Gather statistics about your users. Either you will learn something useful, or at least you'll be able to build some nice graphics from those results (like a map showing where your users are and how the geographical distribution evolves over time).
Analyze the user flow on your website to optimize it (this may be obsolete if you changed the website a lot),
Try to detect which requests were the hacking attempts. Based on those results, can you detect the current hacking attempts faster?
Track the HTTP responses and compare them to your error reporting tool. What you must expect is to find the most 500-type and 404-type requests being reported through the error reporting tool and being analyzed and solved. What you don't want to find is that half of your customers encountered 500/404-type errors regularly on your website, and you were unaware of that and did nothing to solve this.
Example 2: files
Having over 100.000 files on your PC can be another source of large sets of data, especially if you start logging what happens to those files. For example, years ago I did a monitoring system which remembered the size of some directories to alert me when something goes wrong (i.e. large increase in size). The collected data is pretty small, but you can do a similar tool which will collect more detailed data to process it later.
What about, for example, displaying graphically the flow of your data between the directories over time, to show the ones which grow or shrink, the files which are changed, and those which never change? Or, if you enjoy more business-oriented applications, you can process those statistics to decide what files must have more or less backups, or to modify your system by putting the most accessed files to an SSD.
Example 3: log the server activity
What about logging the activity of your server to determine that, for example:
When you are logged in through Remote Desktop but do nothing, the CPU is still used more than when there are no logged in users. Log off to reduce the power usage of your server.
Every Saturday at 3 AM, there is a peak of CPU usage that you don't expected. Maybe there is a scheduled task that you don't need?
For the last four days, the average memory usage increased by 10 MB, while you haven't changed anything on your server. How to explain that?
For the last two hours, the CPU usage is constantly at 95%, while it was at 25% before. Is there a DDOS attack?
All those scenarios are not really Big Data cases, since you deal with a few GB of data, not tera, exa or petabytes. But still, if you want to do something which is:
much more invigorating than just creating yet another e-commerce or web portal for a commercial company
you can play with the data you have right now.