Supposedly Feynman once said the way to look like a genius is to constantly keep several unrelated problems in your head and try to use each new idea or piece of knowledge you acquire with each one of them. Chances are you will figure one of them out eventually, usually when people are not expecting it. This is my attempt at looking smart 
Other sources of ideas
As usual people much smarter then me have their own lists of interesting problems, here some of them. Derek Abott collects his imponderables, a mixed bag of topics there should be somethign for everyone. Mainly about quantum mechanics, Christian Fuchs collects essays here.
If you ever run out of things to read, look at this list. If you manage to finish it let me know, I think it is impossible.
Problems
Archiving data (long term)
Considering that CDs are only good for about 20years and hard disks fail even more quickly, how would you go about storing the vast amounts of data produced daily by researchers? Already data from space missions that took place in the 1970s is not accessible, because we do not have the hardware from those days any more. The experiments at the Large_Hadron_Collider will produce about 1PB of data during each year of running. Just storing all of this data is already a challenge, analysing it will take even longer and making it available to future generations is probably just as big a challenge as finding Higgs.
In order to archive the data you not only have to keep the data itself but also the software used to create it, examples of software to read it and give meaning to the individual bits&bytes.
Some google results:
Mark on Long-term backup
Stanford is building a vault http://infolab.stanford.edu/~cooperb/pubs/ia.pdf
A charge in free fall
Will you see a charge radiate in a frame of reference attached to the charge when it is falling(accelerating) towards earth? ED in non inertial frames might help to answer this.
Old and elsewhere
Ideas that have their own page in this wiki.
Arduino -- a little "labview card". It would go well with one of the Toys
Venti is the Plan9 idea of a backup system. They certainly have a point with "do not delete anything", ZFS is another one of these file systems that are totally different.
Gobby -- can we use something like this to take notes together during a lecture? It seems lecture notes are a very individual thing.
Reading -- a not up-to-date list of books I have recently read.
MapReduce, the now famous google algorithm. While you are over there also have a look at GFS.