Artemisia, like her fourth century name-sake, was not only an able ruler but one of the few female admirals. They were also both accepted as Satrap by the Persian court despite there being adult men in the line of succession [Suda A 4030]:
Ἀρτεμισία: αὕτη ἠρίστευσε κατὰ Περσῶν: δι' ἣν εἶπεν ὁ βασιλεὺς, ὡς οἱ ἄνδρες γυναῖκες γεγόνασιν, αἱ δὲ γυναῖκες ἄνδρες. Ἀρτεμισίαι δὲ δύο γεγόνασι, Καρικαὶ γένος καὶ βασιλίδες ἀμφότεραι, ὧν ἡ μὲν πρώτη γέγονε κατὰ Περσίδα, ἡ δὲ νεωτέρα, ἧς καὶ Δημοσθένης ἐν τῇ περὶ τῆς Ῥοδίων ἐλευθερίας μνημονεύει, θυγάτηρ μὲν ἦν Ἑκατόμνου, γυνὴ δὲ καὶ ἀδελφὴ Μαυσώλου.
This woman was outstanding in serving Persians; because of her the King said that the men had become women and the women men. Two Artemisias existed, Carian by nationality and queens both. The first of them lived in the Persian period; the younger, of whom Demosthenes makes mention in On the freedom of the Rhodians, was daughter of Hekatomnos and both wife and sister of Mausolus.
The major source for Artemisia's life is Herodotus. The historian was born in Halicarnassus, and grew up hearing of her great deeds from men who had served under her in battle. As well as his own Histories, we learn a few facts about him from the Suda. Herodotus, though a fan of Artemisia, was not so keen on her grand-son [E 536]:
Ἡρόδοτος, Λύξου καὶ Δρυοῦς, Ἁλικαρνασεύς, τῶν ἐπιφανῶν, καὶ ἀδελφὸν ἐσχηκὼς Θεόδωρον. μετέστη δ' ἐν Σάμῳ διὰ Λύγδαμιν τὸν ἀπὸ Ἀρτεμισίας τρίτον τύραννον γενόμενον Ἁλικαρνασσοῦ: Πισίνδηλις γὰρ ἦν υἱὸς Ἀρτεμισίας, τοῦ δὲ Πισινδήλιδος Λύγδαμις. ἐν οὖν τῇ Σάμῳ καὶ τὴν Ἰάδα ἠσκήθη διάλεκτον καὶ ἔγραψεν ἱστορίαν ἐν βιβλίοις θ#, ἀρξάμενος ἀπὸ Κύρου τοῦ Πέρσου καὶ Κανδαύλου τοῦ Λυδῶν βασιλέως. ἐλθὼν δὲ εἰς Ἁλικαρνασσὸν καὶ τὸν τύραννον ἐξελάσας, ἐπειδὴ ὕστερον εἶδεν ἑαυτὸν φθονούμενον ὑπὸ τῶν πολιτῶν, εἰς τὸ Θούριον ἀποικιζόμενον ὑπὸ Ἀθηναίων ἐθελοντὴς ἦλθε κἀκεῖ τελευτήσας ἐπὶ τῆς ἀγορᾶς τέθαπται. τινὲς δὲ ἐν Πέλλαις αὐτὸν τελευτῆσαί φασιν. ἐπιγράφονται δὲ οἱ λόγοι αὐτοῦ Μοῦσαι.
Son of Lyxus and Dryo; of Halicarnassus, from a prominent family; he had a brother Theodorus. He migrated to Samos of Lygdamis, who was the third tyrant of Halicarnassus after Artemisia: Pisindelis was the son of Artemisia, and Lygdamis the son of Pisindelis. In Samos practised the Ionian dialect and wrote a history in nine books, beginning with Cyrus the Persian and Candaules the king of the Lydians. He went back to Halicarnassus and drove out the tyrant; but later, when he saw that the citizens were jealous of him, he went of his own will to Thurii, which was colonized by the Athenians, and after he died there, was buried in the agora. But some say that he died in Pella. His Books bear the inscription of the Muses.
There is also a cryptic reference in the Suda [P 1551] to a Pigres, brother of Artemisia the wife of Mausolus:
Πίγρης, Κὰρ ἀπὸ Ἁλικαρνασοῦ, ἀδελφὸς Ἀρτεμισίας, τῆς ἐν τοῖς πολέμοις διαφανοῦς, Μαυσώλου γυναικός: ὃς τῇ Ἰλιάδι παρενέβαλε κατὰ στίχον ἐλεγεῖον, οὕτω γράψας: μῆνιν ἄειδε, θεά, Πηληϊάδεω Ἀχιλῆος, οὐλομένην. μοῦσα, σὺ γὰρ πάσης πείρατ' ἔχεις σοφίης. ἔγραψε καὶ τὸν εἰς Ὅμηρον ἀναφερόμενον Μαργίτην καὶ Βατραχομυομαχίαν.
A Carian from Halicarnassus; brother of Artemisia, the woman of military renown, wife of Mausolos. Pigres was the one who inserted elegiac lines into the Iliad, writing thus: 'sing, goddess, of the ruinous wrath of Achilles, son of Peleus. Muse, for you possess the means of all wisdom'. He also wrote the Margites and Battle of Frogs and Mice attributed to Homer.
The Batrachomyomachia may in fact be as late as the Roman period. Idreus and Pixodorus are the only known brothers of Artemisia and Mausolus. The assumption is that Pigres was the brother of Artemisia I of Halicarnassus, which raises an interesting point; we know that she was chosen to rule instead of her son, the assumption being that the son was too young. If she also had a brother, why was he not made Satrap?
This and other obscure points will be taking up much of my time, so posting might be a little light.
Artemisia was an extraordinary woman, and well worth reading up about - some extracts from Herodotus about her life can be found here.