London Ed of Beyond Necessity does a good job patiently explaining the 'morning star' - 'evening star' example to one of his uncomprehending readers. But I don't think Ed gets it exactly right. I quibble with the following:
(1) The sentence “the morning star is the evening star” has informational content.
(2) The sentence “the morning star is the morning star” does not have informational content.
(3) Therefore, the term “the morning star” does not have the same informational content as “the evening star”.
One quibble is this. Granted, the two sentences differ in cognitive value, Erkenntniswert. (See "On Sense and Reference" first paragraph.) The one sentence expresses a truth of logic, and thus a truth knowable a priori. The other sentence expresses a factual truth of astronomy, one knowable only a posteriori. But note that Frege says that they differ in cognitive value, not that the one has it while the other doesn't. Ed says that the one has it while the other doesn't -- assuming Ed is using 'informational content' to translate Erkenntniswert. There is some annoying slippage here.
More importantly, I don't see how cognitive value/informational content can be had by such subsentential items as 'morning star' and 'evening star.' Thus I question the validity of the inference from (1) & (2) to (3). Neither term gives us any information. So it cannot be that they differ in the information they give. Nor can they be contrasted in point of giving or not giving information. Information is conveyable only by sentences or propositions.
I say this: neither of the names Morgenstern (Phosphorus) or Abendstern (Hesperus) have cognitive value or informational content. (The same holds, I think, if they are not proper names but definite descriptions.) Only indicative sentences (Saetze) and the propositions (Gedanken) they express have such value or content. As I see it, for Frege, names have sense (Sinn) and reference (Bedeutung), and they may conjure up subjective ideas (Vorstellungen) in the minds of their users. But no name has cognitive value. Sentences and propositions, however, have sense, reference, and cognitive value. Interestingly, concept-words (Begriffswoerter) or predicates also have sense and reference, but no cognitive value.
I also think Ed misrepresents the Compositionality Principle. Frege is committed to compositionality of sense (Sinn), not compositionality of informational content/cognitive value. So adding the C. P. to his premise set will not validate the above inference.