Yes Imrahil's family were the greatest nobles in Gondor. For example in note 244 it is said:
Also to be Prince of Ithilien, the greatest noble after Dol Amroth in the revived Númenórean state of Gondor, soon to be of imperial power and prestige, was not a 'market-garden job' as you term it. [The Letters of J.R.R. Tolkien]
They were kinsmen of Elendil and he installed them as Princes, or perhaps they were installed as such by the Steward Mardil. The first Lord of Dol Amroth in that case. These two traditions are mentioned in Unfinished Tales.
It was given to his ancestors by Elendil, with whom they had kinship. They were a family of the Faithful who had sailed from Númenor before the Downfall and had settled in the lands of Belfalas … the first Lord of Dol Amroth was Galador (c. Third Age 2004-2129), the son of Imrazôr the Númenórean, who dwelt in Belfalas, and the Elven-lady Mithrellas, one of the companions of Nimrodel … the two statements can only be reconciled on the supposition that the line of the Princes, and indeed the place of their dwelling, went back more than two thousand years before Galador's day, and that Galador was called the first Lord of Dol Amroth because it was not until his time (after the drowning of Amroth in the year 1981) that Dol Amroth was so named.
Based on these two traditions they first became lords of Dol Amroth only after there was such a name, Dol Amroth, to be had, but were installed as Princes of that land by their kinsman Elendil 2000 years earlier.
Speaking of Elendil, keep in mind in Númenor his house was likewise the greatest nobility in the land after the king's house. When the last king died they took up the mantle.
The Stewards were a stand in for the Kings. The only option after this would be the Prince, like the Princes did in Cardolan after their Kings died out. Back to note 244 it states:
The chief commanders, under the King, would be Faramir and Imrahil; and one of these would normally be a military commander at home in the King's absence.
You can again see the primacy of the Princes after the Kings.
Let me add in this edit one more passage from The People of Middle-earth. The Stewardship itself could pass in the female line;
if a Steward left no son, the office might pass in the female line, that is to his sister-son, or to his father's sister-son.
The choice was made according to their worth among the near kin by the Council of Gondor. But the Council had no power of choice if there was a son living.
As you can see if there was no son, it could pass to a nephew or a cousin in the female line. See for example the 10th Steward Denether l.