A few possibilities come to my mind.

Looking at the overall hit rate is usually not a very good idea as it will depend on the composition of the test set if the performance for the different classes differs. So at the very least, you should be specify (and justify) the relative frequency of the classes in your test data in order to derive a meaningful value.

Secondly, as @Shorack already said, specify which types of error are how important. Often, the classifier needs to meet certain performance criteria in order to be useful (and overall accuracy is rarely the adequate measure). There are measures like sensitivity, specificity, positive and negative precdictive value that take into account the different classes and different types of misclassification. You can say that these measures answer different questions about the classifier:

• sensitivity: What fraction of cases truely belonging to class C is recognized as such?
• specificity: What fraction of cases truely not belonging to class C is recognized as such?
• positive predictive value: Given the classifier predicts class C, what is the probability that this prediction is correct?
• negative predictive value: Given the classifier predicts that the case is not form class C, what is the probability that this prediction is correct?

These questions often allow to formulate specifications that the classifier must need in order to be useful.

The predictive values are often more important from the point of view of the practical application of the classifier: they are conditioned on the prediction, which is the situation you are in when applying the classifer (a patient usually is not interested in knowing how likely the test is to recognize diseased cases, but rather how likely the stated diagnosis is correct). However, in order to properly calculate them you need to know the relative frequencies of the different classes in the population the classifier is used for (seems you have this information - so there's nothing that keeps you from looking at that).

You can also look at the information gain that a positive or negative prediction gives you. This is measured by positive and negative likelihood ratio , LR⁺ and LR⁻. Briefly, they tell you how much the prediction changes the odds towards the class in question. (see my answer here for a more detailed explanation)

For your trivial classifier, things look like this: I'll use the "0" class as the class in question, so "positive" means class "0". Out of 100 cases, 100 are predicted positive (to belong to class 0). 97 of them really do, 3 don't. The sensitivity for class 0 is 100% (all 97 cases truely belonging to class 0 were recognized), specificity is 0 (none of the other cases were recognized). positive predicitve value (assuming the 97:3 relative frequency is representative) is 97%, negative predictive value cannot be calculated as no negative prediction occurred.

$LR^+ = \frac{\text{sensitivity}}{1 - \text{specificity}} = 1$
$LR^- = \frac{1 - \text{sensitivity}}{\text{specificity}} = \frac{0}{0}$
Now LR⁺ and LR⁻ are factors with which you multiply the odds for the case to belong to the positive class ("0"). Having an LR⁺ of 1 means that the positive prediction did not give you any information: it will not change the odds. So here you have a measure that clearly expresses the fact that your trivial classifier does not add any information.

Completely different direction of thoughts: You mention that you'd like to evaluate different classifiers. That sounds a bit like classifier comparison or selection. The caveat with the measures I discuss above is that they are subject to very high random uncertainty (meaning you need lots of test cases) if you evaluate them on "hard" class labels. If your prediction is primarily continuous (metric, e.g. posterior probability) you can use related measures that look at the same kind of question but do not use fractions of cases but continuous measures, see here. These will also be better suited to detect small differences in the predictions.

(@FrankHarrell will tell you that you need "proper scoring rules", so that is another search term to keep in mind.)