Just like the title said.
What a heir to a hereditary Duchy is called? (as opposed to someone appointed by a King). I mean a Duchy where the Duke inherited his title to his son/daughter.
Since the only sovereign Duchy in the world is Grand Duchy of Luxembourg where they styled the heir as "Hereditary Grand Duke", then is by the same analogy, the heir of a hereditary Duchy is "Hereditary Duke"?
The children of a sovereign Grand Duke may be titled "Prince" (Luxembourg, Tuscany, Baden, Hesse-Darmstadt, Saxe-Weimar) or "Duke" (Oldenburg) in accordance with the customs of the dynasty. The heir of the throne of a Grand Duchy is titled "Hereditary Grand Duke", as soon as he reaches the full legal age (majority). wikipedia
I continue to maintain that the real answer is, "Whatever the currently reigning Archduke says to call him", or "Whatever tradition says" - but wikipedia's (unsourced) assertion has some merit. If the ruler is sovereign then the heir apparent is a Prince.
Update: As @Semaphore has pointed out, Grand Dukes are distinct from Archdukes. The critical fact is that the duke is sovereign. A sovereign ruler use any title including Duke, Grand Duke, ArchDuke or "poobah and bottle washer"; the heir apparent to a sovereign rule is a "Prince" unless local tradition or autocratic decree mandates a different title.
Don't be misled by translations and cognates (words in different languages sharing a common root). German "mist" and English "mist" are cognates - but German "mist" is what farmers spread on their fields at night. Dignities are what they are, and reflect the specific history of every family - translation can only attempt description that is non-offending to title holders.
The Habspurg royal family has never used the title Archduke - that is an English-language description of the German title Erzherzog that attempts to place it into an English title system without insulting the dignity of any holders. The plain-language translation of "erz" from German to English is "ore" not "arch".
German Prinz(essin) are cognate to English "Prince(ss)" - but that doesn't make them the same. The Prince(sse)s Electors of the Holy Roman Empire were titled "Kurfürst(in)" and not Prinz(essin). Notwithstanding that "Kronprinz(essin)" and "Crown Prince" happen to have the same meaning, in German a ruling "prince(ss)" is most commonly a "Reichsfürst(in)", not of a "principality" but rather of a "Fürstentum".
The style of a sovereign is very much a political statement - in both formal and informal contexts. The style of British sovereigns included a claim to the French throne almost continuously from 1340 through 1801.
All claims contrary to the above are, frankly, mist - in the German sense.
Crown Prince is a colloquial (and descriptive) term, not a proper title. The phrase Heir Apparent likewise is simply a descriptive term rather than a title. The official title for various European royal heirs are:
For British nobility (as opposed to royalty above) it is traditional for the heir of a peer to hold a courtesy title selected from the lesser titles held by the peer. In the case of a Duke this is most often a title of marquess, but may be earl or viscount depending on the lesser titles available. The heir of an heir in turn may be granted a courtesy title (from those available) at least one step down from his parent, and so on, until only Lord is available. Note that while holding only a courtesy title, such an heir would be eligible to sit in the House of Commons, not (yet) being a Peer eligible (only) to sit in the House of Lords.
For German nobility it was common for a nobles heir to prefix his parent's title with "Erb", as in "Erbgraf". William Addams Reitwiesner writes:
"Erb" in German (in this sense) means "hereditary"[. . . .] The oldest son and heir of a Mediatized Count would be an "Erbgraf". The oldest son and heir of a Grand Duke would be an "Erbgroßherzog". And so on. Another way of spelling the title would be "Erb-Prinz" or "Erb-Graf", etc. The wives of these men have equivalent feminine titles, such as"Erbprinzessin", "Erbgräfin", "Erbgroßherzogin", etc. The French form is "prince heredetaire", "comte heredetaire" "grand-duc heredetaire", etc. (toss in accents as appropriate).
In Europe some titles descend by primogeniture, and some titles descend to all agnatic (male line) descendants of the first title holder. And the same goes for hereditary rulers - in some principalities the rule descended to one person by primogeniture, but in other principalities the land divided among the sons of a ruler upon his death, so those principalities became smaller and smaller and more and more numerous over the years.
In the Lichtenstein dynasty, for example, the title of prince is shared by all members of the dynasty, but the rule of the principality descends to one person by agnatic primogeniture. So there are many princes of the Lichtenstein dynasty at any time but only one ruling and reigning prince of Lichtenstein at a time.
Many titles are granted so a higher title goes by primogeniture and a lower title goes to all agnatic members of the grantee's descendants. So the dynasty might have one duke and several lower ranking princes (or vice versa) or it might have one duke at a time and several counts, and so on in various other combinations.
As nearly as I can tell, it is common to describe the heir to the Duchy of X or Principality of Y as the hereditary duke of X or the Hereditary Prince of Y.
In the UK it is common for a noble to have a number of different titles. The Duke of A might be in full Duke of A, Marquess of B, Earl of C and D, Viscount of E, and Baron of F, G, H, and I, for example. So it is common to call a Duke's eldest son and heir apparent by the next senior title the Duke has - but that is just a courtesy title. So the Duke of A's heir could be called by the courtesy title of Marquess of B. And I guess the Duke's senior grandson could be called the Earl of C, and his senior great grandson could be called the Viscount of E, and his senior great great grandson could be called the Baron of F.
PS missing a letter from my keyboard.
added 03-20-2020 I put in the missing "w"s.
In a Sovereign duchy the title is either a specialized title of nobility, Count of N or the like, hereditary duke or prince. The heir of a non-reigning duke is Lord N, a lesser one of his/her father/mother's titles, or no title at all if he is a distant relative.
For example, the Grand Duke of Tuscany's Heir is the Grand Prince of Tuscany.
