Ok, let's go through this http://www.news.com/8301-10784_3-9932054-7.html
"The fourth circuit, which Chua called a "memristor" for memory resistor, would register how much current had passed."
Firstly, current is a measure of the flow of charge so you are actually measuring how much charge has passed.
It seems that people really don't get electricity at all. It's like people saying that the power companies provide current. They don't, they provide voltage and you the customer draw the current you required. The best I've heard (on this site even) is people saying that the power grid provids 120v of current!
Secondly, isn't this what a capacitor does? As charge accumulates on the plates of a capacitor (assuming your basic 2 plate model) then a potential difference is generated that is proportional to the charge. By measuring the voltage on the capacitor you know how much charge has passed, assuming that you are looking at a circuit whose initial conditions are known.
"a memristor is a variable resistor that, through its resistance, reflects its own history, Williams said."
No it doesn't. It provides an instaneous value of its resistance without any connection to previous (or future) values. If you apply a constantly varying voltage across the device you will be unable to gain any insight into the variation that occurred prior to your measurement, all you get is an average.
Other than that, nothing really in it. It is interesting that the dicussion on Wikipedia for this subject has pointed out that most material on this topic has appeared very recently and are mostly written by journos not 'experts'.