I assume you are not planning to give your son a name in Latin, but merely a name from Latin. When used as names outside of a Latin-language context, the forms Amadeus, Amadeo and Amédée should all have equivalent meanings. The first is Latinate in form, the second Italian, and the third French. Due to phonetic sound changes between Latin and the Romance languages, the Latin nominative suffix -us corresponds to Italian -o. I don't really understand why the French form has a final mute "e", but that isn't related to inflected forms either: French -ée just corresponds to Latin -eus in many names (like Persée "Perseus", Thésée "Theseus", etc.)
Amadei is the genitive form of Amadeus in Latin. That is, it can mean more or less "of Amadeus" (or in Italian, "of Amadeo"). This sense naturally gives rise to the sense "son of Amadeo" when applied to a person; compare the example of the famous Galileo Galilei, whose name Wikipedia explains as follows:
The surname Galilei derives from the given name of an ancestor, Galileo Bonaiuti, a physician, university teacher and politician who lived in Florence from 1370 to 1450; his descendents had changed their family name from Bonaiuti (or Buonaiuti) to Galilei in his honor in the late 14th century. [...]
Latin inflection of "Amadeus"
In Latin, the name "Amadeus" would be inflected into different forms like "Amadeum", "Amadeo" etc. depending on the grammatical context, but inflected forms like this are not normally used as the basis for names in other languages. For example, the Latin name Marcus has the vocative form "Marce", but this is not commonly used as a name in English.
In any case, these forms would not change the understood meaning of the name. They would change how sentences containing the name would be interpreted.
The vocative of "deus" in Latin is actually complicated; it's often said to be "deus" instead of "dee" (although I have the impression that the latter form does occur sometimes). Actually, though, I think that you are right about Amadee being the vocative of Amadeus, because, assuming the name does in fact mean "love God" (something I'm not entirely sure of, despite the claims of Wikipedia), I can see no logical reason for it to contain the word "God" in the nominative case. (In the sentence "love God", "God" is the object of the sentence.)
Therefore, it seems to me that the nominative suffix -us at the end of the word should not be considered to form the word "deus" along with the preceding "de", but rather, it should be seen as a suffix attached to the entire preceding word: it looks to me like it would be divided [Ama-de]-us, with the [Ama de] part coming from a truncation of "Ama deum" ("Love God") and the -us acting as a suffix to convert this phrase to a masculine name. I'm not certain that I'm right about this, but this kind of analysis makes the most sense to me provided we stipulate that the name means "Love God!", and it implies that the inflection in other case-forms should be like that of a normal second-declension noun rather than like that of "deus".
In any case, I found a post on the Textkit forum that does say "Amadee" is a correct vocative form:
Perseu est Perseus vocativo casu quod vocabulum Graecum.
Amadee est Amadeus quod Latinum seu Italicum.
Deus est Deus vocativo.
–adrianus » Thu Sep 12, 2013 7:21 pm
I also saw a WordReference post that refers to the name as "dog-Latin", so it may be a bit pointless to try to interpret it as a correctly formed Latin compound in any case – as ktm5124 points out, it is meant to parallel formations in other languages like Greek Θεόφιλος and German Gottlieb.