Thought 1: Building an electrical model of a neuron.
A widely used engineering model for a neuron is the McCulloch and Pitts model. There are others, but none seem to be a that good a model for a real neuron.
Can you review the current state of the art and come up with or improve on an alternative electrical circuit model for a neuron that you can build and test electrically. Is it possible to build one that shows a much greater resemblance to the properties of a real neuron, than the McCulloch and Pitts model?
See for example
"An Introduction to Neural Computing", by Igor Aleksander and Helen Morton
"The Computational Brain", Patricia S. Churchland and Terrance J. Sejnowski
If you are not interested in investigating biological analogies then there are plenty of more mainstream topics.
Thought 2: One of the interesting things I used to play with is putting a magnet on an electron beam oscilloscope.
What you get is a sort of quick visual model for how charged particles might distribute themselves on hitting the earth's magnetic field. By taking digital photographs, try to map and quantify the distributions of light you get on the oscilloscope screen as you get as you move the magnet across the oscilloscope screen. Then try to model this and see if this model agrees with our understanding of what happens when charged particles from the solar wind become influenced by the earths magnetic field.
Thought 3: There are certain oscillator circuits using thermistors that can produce sine waves with low harmonic distortion and noise. What are the physical limits to how perfect you can make a sine wave output from an electronic oscillator circuit? Once you have created your oscillator circuit can you find a way of measuring its imperfections?
These thoughts are completely unrelated. I could have given a thought 4 in relation to thought 1, but I am sure you can come up with better ideas than me if you really force yourself to think creatively.