What were the heir and children of monarchs with the titles "Grand Prince" or "Prince" called?

Short answer: I don't know.

If any sons or other relatives of Medieval Eastern European princes had their own land to rule they would be known by the title of the ruler of that province. And perhaps sometimes the other sons and relatives of the prince didn't have any other titles. And probably there were various titles used for the sons and other relatives of the ruler in different times and places.

But I am not familiar with those titles which may have been used.

Long Answer.

Prince is a word in the English language. It has several different meanings.

In one meaning a prince can be any ruler of any rank.

In a second meaning a prince is a ruler who has the title of prince and rules a principality which is a dependent part of a larger country, or in two modern cases, Monaco and Liechtenstein, is an independent country.

In a third meaning a prince is a noble who has the title of prince but doesn't rule anywhere. In some systems of nobility a prince in the second or third meaning ranks higher than a duke, while in other systems of nobility a duke ranks higher than a prince.

In a fourth meaning a prince is a son or other close relative of a king, a royal prince. Royal princes rank higher than noble princes or dukes.

In a fifth meaning, a prince is the ruler of a principality as in the second meaning, but has a different title, like landgrave or duke.

In a sixth meaning a prince is a son or other close relative of a duke or a prince in the second, third, or fifth meaning, and has a title of prince or is called a prince.

And there are other meanings of prince which have nothing to do with rulers or royalty.

And the words in many languages which get translated into English as prince, and that prince gets translated into, often also have several different meanings.

In German there are two words that are translated as prince, which have about three meanings.

The English word prince is ultimately derived from the Latin princeps, meaning "first". Early Roman emperors often used princeps as a title, since they were the "first" and most powerful men in the Roman republic and empire.

The German word Fürst is derived from the German word for the first, and so analogous to the Latin word princeps, and so is used to mean a prince in the second, third, and fifth meanings of the English word prince.

The princes of the Holy Roman Empire, or Fürsten, were each the first man in their principality and together were the first men in the Empire.

And most of them were princes in the fifth English meaning, having titles other than Fürst. Titles of German ruling princes from the lowest to the highest were princely count, landgrave, margrave, count palatine, prince itself, and duke. And there was one grand duke and one even higher archduke.

Emperors also granted titles without fiefs or principalities. Those titles went from noble, baron, count, and prince, to duke.

And there were rules, laws, and customs about which members of a family could share the title of the head of the family. In some cases the junior members of a dynasty confusingly used the same title as the head of the family, and in other cases they had to use other and lower titles.

The other German world translated as prince is Prinz. That means a prince in the fourth English meaning, a son or close relative of a king. And I think that it probably also means a prince in the sixth English meaning a son or close relative of a prince in the sense of a Fürst.

Of course the Holy Roman Empire is usually considered to have been in central Europe, not Eastern Europe.

Most of Eastern Europe, with notable exceptions like Finland, the Baltic countries, Rumania, Hungary, and Greece, speaks various Slavic languages.

The oldest Slavic words for prince were variations of knyaz, but many languages later adopted words meaning "son of the king" or derived from princeps and Prinz. And I am not certain whether it would be more accurate to translate knyaz and similar words as "duke", or "prince", or "king", or something else.

And in the Middle Ages vast regions of Eastern Europe first became united under a single ruler and later became divided into many subordinate states with large degrees of autonomy, whose rulers are described as "dukes" or "princes".

There were several dukes or princes in Poland under a senior duke (or senior prince) or a king a different times. In the 14th century Lithuania conquered a vast region, becoming the largest country in Europe for a time. And there were many duchies or principalities in what later became Russia.

"Grand duke" is the traditional translation of the title Velikiy Kniaz (Великий Князь), which from the 11th century was at first the title of the leading Prince (Kniaz) of Kievan Rus', then of several princes of the Rus'. From 1328 the Velikii Kniaz of Muscovy appeared as the grand duke for "all of Russia" until Ivan IV of Russia in 1547 was crowned as tsar.

Velikiy knyaz (Meaning closest to Grand Prince but was generally translated as Grand Duke in state documents written in Latin), used in the Slavic and Baltic languages, was the title of a medieval monarch who headed a more-or-less loose confederation whose constituent parts were ruled by lesser knyazs ( often translated as "princes" ) . Those great knyazs' (grand princes') title and position was at the time sometimes translated as king, though kings, princes, and dukes seemingly initially did not exist amongst proto-Slavs and Balts with Knyaz being a Germanic loanword adopted by tribal chieftains. Although, the Slavic knjaz and the Baltic kunigaikštis (nowadays usually translated as prince) are similar to kings in terms of ruling and duties. However, a velikiy knyaz (grand prince) was usually only primus inter pares within a dynasty, primogeniture not governing the order of succession. All knyazs (princes) of the family were equally eligible to inherit a crown (for example, succession might be through agnatic seniority or rotation). Often other members of the dynasty ruled some constituent parts of the monarchy/country. An established use of the title was in the Kievan Rus' and in the Grand Duchy of Lithuania (from the 14th century). Thus, Veliki Knjaz has been more like a regional high king (but without international recognition as such) than "grand duke", at least, originally and were not subordinated to any other authority as more western (for example Polish) Grand Dukes were.[citation needed] As these countries expanded territorially and moved towards primogeniture and centralization, their rulers acquired more elevated titles.

Великий князь (Velikiy Knyaz; literally, great prince) was, starting in the 10th century, the title of the leading Prince of the Kievan Rus', head of the Rurikid House: first the prince of Kiev, and then that of Vladimir and Galicia-Volhynia starting in the 13th century. Later, several princes of nationally important cities, which comprised vassal appanage principalities, held this title (Grand Prince of Moscow, Tver', Yaroslavl', Ryazan', Smolensk, etc.). From 1328 the Grand Prince of Moscow appeared as the titular head of eastern Rus' and slowly centralized power until Ivan IV was crowned tsar in 1547. Since then, the title grand prince ceased to be a hereditary office and became a generic title for members of the Imperial family until the Russian Revolution of 1917.

The Lithuanian title Didysis kunigaikštis was used by the rulers of Lithuania, and after 1569, it was one of two main titles used by the monarch of the Polish–Lithuanian Commonwealth. The kings of Poland from the Swedish House of Vasa also used this title for their non-Polish territories. This Lithuanian title was sometimes latinized as Magnus Dux or Grand Duke.

Any son or other relative of the senior duke, or high duke, or the king in other eras, of Poland, who had a duchy of his own would be a duke. But I don't know what the members of he Piast dynasty, sons and other relatives of the ruler of all Poland, were called when they didn't have a duchy to rule as a vassal.

Any son or other relative of the Grand Duke or Great Prince of Lithuania who had a duchy or principality of their own to rule was a duke or prince. But I don't know the members of the Gediminid dynasty, sons and other relatives of the grand duke of Lithuania, were called when they didn't have any duchy or principality of their own.

Any son or other relative of the Grand Duke or Great Prince of Russia who had a duchy or principality of their own to rule was a duke or prince. But I don't know what the members of the Rurik dynasty, sons and other relatives of the grand duke of Russia, were called when they didn't have any duchy or principality of their own.

There is unlikely to be one answer which applies to the whole breadth of time and place that you've included; you'd have to focus on a particular state or tradition. Very few countries in the world have officially used English titles, so the question of translation also comes into play: whose authority would we accept that a particular Eastern European title should be translated as "Prince" rather than "King"?

However, there are some languages which use a very similar word to the English "Prince", including French, which has a word spelled identically. Furthermore, we don't have to go back into history to find a ruler bearing that title: meet Albert II, le prince souverain de Monaco.

So, we can answer in at least one case: the heir to the Principality of Monaco, currently Albert's son Jacques, is styled le prince héréditaire de Monaco, which directly translated is "hereditary prince of Monaco".

Liechtenstein is also generally called a "Principality" in English, although the title in German is "Fürst", not the more obviously cognate "Prinz", which is used as a lower ranking title. The heir to the throne bears the German title "Erbprinz". "Fürst" and "Erbprinz" are generally translated to English as "Prince" and "Hereditary Prince", respectively.

There is also the Principality of Andorra, which has joint rulers bearing the title (in Catalan) "Copríncep", but neither have heirs - at any time, they are the current Bishop of Urgell, and the current President of France. Although particularly unusual, this shows how hard it is to make any general rule about what titles would be applied to whom.