I am not sure I understand the problem. C is related to both answer and u_p. The two models are:

logit(answer_ip) = beta0 + u_i + u_p,

logit(answer_ip) = beta0 + beta1 C_p + u_i + u_p,

C_p accounts for some of the variation in answering for the people, and therefore u_p changes. Is the question about the meaning of u_p changing depending on what other variables are included in the model?

Thank you, Daniel. That's exactely what I meant (if you identify u_p in the lower model by participant ability, i.e., u_p + beta1*C_p).

u_p in the upper model represents the estimate of participant ability that is supposed to comprise the variance bound by C_p. A subsequent regression of the conditional means of u_p on C_p results in a weaker beta1 as compared to the lower model. Such two-step procedures seem to be quite common in personality psychology.

In the lower model u_p (+ beta1 C_p) increases the dispersion of participant ability (u_p in in the upper model) while preserving its rank order, which finally results in a larger estimate of beta1. This result can be confirmed by applying a subsequent regression as outlined above (which is kind of self-evident as u_p = beta1 C_p + residuals by definition)

In consequence, the same covariate may not significantly predict participant ability in the upper model, whereas in the lower model it may actually have a predictive value. Thus both approaches may suggest completely diametral interpretations, although they measure the same quantity (participant ability).

I wouldn't worry about statistical significance. In the example you gave, C accounts for some of person ability in the top model, and some of the response accuracy in the bottom. Depending on how the computer has done the estimation, the u_p may differ. They (what statisticians call the condition modes, but which people often refer to as the person residuals) are estimates that for most models borrow information from each other. This means some accuracy values that are very different from the others will be estimated as not as different. They are often called shrunken estimates, and a great paper on it is http://statweb.stanford.edu/~ckirby/brad/other/Article1977.pdf (it has a baseball example, and it is the playoff season here in the US, go Dodgers). If ignoring significance it looks like they tell the same story, that would be fine.

This doesn't get exactly to your question about what to do, but that will depend on your purpose. Do you want to measure ability, or ability after conditioning on something.