In the Odyssey, Athena warns Telemachus to be quickly on his way:

Od. 15. 16-18

“Already [Penelope’s] father and relatives are urging her
To marry Eurymakhos, for he excels all the other suitors
In gifts and he has promised many bridegifts.”

ἤδη γάρ ῥα πατήρ τε κασίγνητοί τε κέλονται
Εὐρυμάχῳ γήμασθαι· ὁ γὰρ περιβάλλει ἅπαντας
μνηστῆρας δώροισι καὶ ἐξώφελλεν ἔεδνα·

The word kasignêtoi here can merely mean male relatives (it is cognate with the English “cousins”) but the Scholia to the Odyssey take the passage to be referring to Penelope’s father and brothers. It names them.

Schol ad. Od. 15.16-17

“Penelope had two brothers, Sêmos and Aulêtês. But Ikarios was from Kephallanian Messêne. He was not seen in Ithaka because he was traveling. But he is not Laconian. This is the reason Telemakhos did not visit him during his trip abroad in Sparta.”

ἀδελφοὶ τῆς Πηνελόπης δύο, Σῆμος καὶ Αὐλήτης· ὁ δὲ ᾿Ικάριος ἐκ Μεσσήνης ἦν τῆς Κεφαλληνιακῆς· ἐπεὶ οὐχ ὁρᾶται ἐν ᾿Ιθάκῃ ἀναστρεφόμενος. ἀλλ’ οὐδὲ Λάκων· ὅθεν οὐδὲ ἐνέτυχε αὐτῷ Τηλέμαχος ἐν τῇ εἰς Λακεδαίμονα ἀποδημίᾳ. H.

So, this scholion excuses Ikarios by putting him elsewhere. (There is debate in the tradition about whether or not this Ikarios is Tyndareus’ brother. This would put him the Peloponnese, and it troubles scholars that Telemachus does not visit his maternal grandfather.) But what of Penelope’s brothers—did they owe no support to their nephew in his time of trouble? (The brothers are not given these names anywhere else that I know of).

Telemachus takes Athena’s warning to heart, as he says later to Theoklymenos:

Homer, Odyssey 15.518-524

“But I will tell you of another man you might encounter,
Eurymakhos, the shining son of sharp-minded Polyboios,
Whom the Ithakans now look upon the way they would a god.
He is by far the best man remaining and the best
To marry my mother and receive my father’s geras.
But Zeus is the one who knows these things as he rules on high”
Whether or not he will bring about a deadly day for them before a marriage.”

ἀλλά τοι ἄλλον φῶτα πιφαύσκομαι, ὅν κεν ἵκοιο,
Εὐρύμαχον, Πολύβοιο δαΐφρονος ἀγλαὸν υἱόν,
τὸν νῦν ἶσα θεῷ ᾿Ιθακήσιοι εἰσορόωσι·
καὶ γὰρ πολλὸν ἄριστος ἀνὴρ μέμονέν τε μάλιστα
μητέρ’ ἐμὴν γαμέειν καὶ ᾿Οδυσσῆος γέρας ἕξειν.

There are other names and other brothers too, as our friend Carly Silver points out:

Love this. Pseudo-Apollodorus adds on, "Icarius and Periboea, a Naiad nymph had five sons" 1/2

— Carly Silver (@CarlyASilver) March 15, 2017

", Thoas, Damasippus, Imeusimus, Aletes, Perileos,192 and a daughter Penelope, whom Ulysses married."

— Carly Silver (@CarlyASilver) March 15, 2017