Can we truly random choose which finger to raise by counting to 10 in your mind? If a test person is asked to raise one of his 10 fingers at a given signal, is the choice truly random if he counts in his mind from 1 to 10 and associates with each number one of his fingers? The 1 he associates with the left pink, the 2 with his left ring finger, 3 with his left middle finger, etc. So the test person starts to count, the experimenter gives him a signal, after which the test person raises the finger associated with the number he arrived at while counting. Let's suppose the signal is given at a random moment. Is the choice "in sync" with the random signal given?

In other words, if the experimenter records which finger is raised after each of a 1000 signals are given to a single person, will the experimenter see that each finger is raised the same number of times (more or less, because the number of signals given is finite; the more signals, the better you're getting at the real distribution)? So in the case of 1000 signals given to a test person will the experimenter record that each finger is raised about 100 times?

Or does the test person not make a choice at all?
I think your question ultimately is about whether people can or are able to behave like a uniform random number generator drawing numbers from 1 to 10.
I don't think the fact that the participant needs to raise a finger changes things much.

Note that there is a distinction between what people do, and what people are capable of. A lot of research suggests that people are not very good at simulating randomness. That said, one can think of strategies that would allow a person to simulate randomness more effectively (e.g., drawing on sources of random information in the environment, or even applying a formula similar to that used to sequentially generate numbers using random number generators).

The broad question has already been posed on the site: 
How well can a human-generated &quot;random number&quot; be predicted?

I also note that a participant could just report sequentially 1,2,3..., 10, 1, ... This data would show 100 of each number for 1000 signals. Of course, this would exhibit structure that would not be present were each trial independent.