Not A Prince...

[Legolas]

'Prince' and other titles for specific members of the royal family were not used often in Tolkien's works.

Legolas obviously is the Prince of Mirkwood, but in Middle-earth, such a title was not carried through different lands. If he ever was called that, I'd estimate that it would've only been in Mirkwood.

The instances of the word 'prince' being used to refer directly to certain persons are few.

1) The princes of the Noldor. There was obviously a struggle for leadership there, and a majority of The Silmarillion is based on these princes. The title of High King of the Noldor was a great title indeed - the ruler over an entire kindred of Eldar.

High princes were Fëanor and Fingolfin, the elder sons of Finwë, honoured by all in Aman;

Thus spoke Maedhros and Maglor and Celegorm, Curufin and Caranthir, Amrod and Amras, princes of the Noldor;

but Galadriel, the only woman of the Noldor to stand that day tall and valiant among the contending princes,

2) Sindarin princes of Beleriand. Olwë was called a prince because he rose to a ruling position when many of the Teleri went to Aman, and King Thingol returned to Beleriand. This is the case with the Sindarin princes spoke of in Unfinished Tales - explaining how they came to rule other lands after the fall of Doriath. It seems Celeborn is referred to as a prince to explain Dior's marraige to Nimloth and Galadriel's marraige to Celeborn - they were marrying other royalty, not just any elves.

As for themselves, they desired now no other home but the strands of Eldamar, and no other lord than Olwë, prince of Alqualondë.

...their son Dior Eluchíl had to wife Nimloth, kinswoman of Celeborn, prince of Doriath, who was wedded to the Lady Galadriel.

There she met Celeborn, who is here again a Telerin prince, the grandson of Olwë of Alqualondë and thus her close kinsman.

3) A title for a woman whose status had suddenly changed and was surrounded by peasant talk.

At all events, suitors for Ancalimë's hand soon began to appear in Emerië, and not only because of the change in her position, for the fame of her beauty, of her aloofness and disdain, and of the strangeness of her upbringing had run through the land. In that time the people began to speak of her as Emerwen Aranel, the Princess Shepherdess.

4) The Prince of Dol Amroth. A formal title of the ruler there.

5) Prince Faramir, whose status as prince was highlighted because he had snuck off into battle disguised, fell, and was found to be Ondoher's son after his death. His father had ordered him to stay at Minas Tirith as regent.

6) Faramir, speaking to Frodo. He used it to highlight Boromir's status as son of Denethor, the Ruling Steward (whom he thought to be above Aragorn before he was told who Aragorn was).

'No, not because I choose,' answered Frodo. `It does not belong to me. It does not belong to any mortal, great or small; though if any could claim it, it would be Aragorn son of Arathorn, whom I named, the leader of our Company from Moria to Rauros.'
'Why so, and not Boromir, prince of the City that the sons of Elendil founded? '

6) Pippin's title as Prince of the Halfings.

He looked now, had he known it, verily Ernil i Pheriannath, the Prince of the Halflings, that folk had called him; but he felt uncomfortable

7) Faramir, Prince of Ithilien - a title Aragorn appointed to him.

Of those listed, only the princes of the Noldor is a title used repeatedly. The others are used one or two times (except in the case of the leadership positions of Faramir and the Prince of Dol Amroth).

As you may notice, 'prince' wasn't necessarily used to denote the son of a king, but to all of the males in the royal family. It also seems to be a generic royal title given to a ruler that did not exercise great authority or gain reverence as a king would, or rulers who shared said leadership.

The lordship over Mirkwood had been held by Thranduil for quite sometime - he was either the first or second ruler of the realm. There was no reason to point out Legolas' princedom, especially since he never returned to Mirkwood for any extended visits.