First, you need to keep in mind that what one would consider "random" might not be "random" for another person. On one extreme, some scientists believe the brain is operating essentially deterministically (thermal/quantum noise is too small), that is, if you know the "precise" state of the brain and its environment, you can predict its future states for a very long time. Regardless of the truth of that statement, neural signals are effectively random due to our lack of knowledge.

Now, you specifically asked if a neuron can make random responses regardless of synaptic input. Yes, it can.

1. There have been many controlled current injections to neurons. For example, in the figure below, **repeated trials with exactly same current step show slight differences in the spike trains**.

![From Mainen and Sejnowski, Science 1995][1]

2. Thermal fluctuations of channels can cause spontaneous fluctuations in the membrane potential.

3. There are spontaneously firing neurons (no synaptic input necessary). (See https://biology.stackexchange.com/a/15259/460)

Figure from: Mainen, Z. F. and Sejnowski, T. J. (1995). **Reliability of spike timing in neocortical neurons**. *Science*, 268(5216):1503-1506.

  [1]: https://i.sstatic.net/ixnrz.png