The word οἰκεῖοι, which as you note is plural of οἰκεῖος, doesn't have an equivalent that is as concise as the Greek. It means household member, but it takes us 2-4 words in English to express its meaning ("household member" or "member of the household" or just, as in the KJV, "of the household") whereas there is a single word in Greek. Brenton suggests "kindred" and "relations" sometimes in his Septuagint translation (e.g. Proverbs 17:9 LXX), but, as we know, not all in a household may be related by blood.
It appears in 2 other places in the New Testament, but over a dozen times in the Septuagint:
Ἄρα οὖν ὡς καιρὸν ἔχομεν, ἐργαζώμεθα τὸ ἀγαθὸν πρὸς πάντας, μάλιστα δὲ
πρὸς τοὺς οἰκείους τῆς πίστεως.
As we have therefore opportunity, let us do good unto all men,
especially unto them who are of the household of faith.
1 Timothy 5:8
Εἰ δέ τις τῶν ἰδίων καὶ μάλιστα τῶν οἰκείων οὐ προνοεῖ, τὴν πίστιν
ἤρνηται, καὶ ἔστιν ἀπίστου χείρων.
But if any provide not for his own, and specially for those of
his own house, he hath denied the faith, and is worse than an
Proverbs 17:9 LXX
ὃς κρύπτει ἀδικήματα, ζητεῖ φιλίαν, ὃς δὲ μισεῖ κρύπτειν, διίστησιν
φίλους καὶ οἰκείους.
He that conceals injuries seeks love; but he that hates to hide them
separates friends and kindred.
Isaiah 3:6 LXX
ὅτι ἐπιλήμψεται ἄνθρωπος τοῦ ἀδελφοῦ αὐτοῦ ἢ τοῦ οἰκείου τοῦ
πατρὸς αὐτοῦ λέγων Ἱμάτιον ἔχεις, ἀρχηγὸς ἡμῶν γενοῦ, καὶ τὸ βρῶμα
τὸ ἐμὸν ὑπὸ σὲ ἔστω.
For a man shall lay hold of his brother, as one of his father’s
household, saying, Thou hast raiment, be thou our ruler, and let my
meat be under thee.
Isaiah 58:7 LXX
διάθρυπτε πεινῶντι τὸν ἄρτον σου καὶ πτωχοὺς ἀστέγους εἴσαγε εἰς τὸν
οἶκόν σου, ἐὰν ἴδῃς γυμνόν, περίβαλε, καὶ ἀπὸ τῶν οἰκείων τοῦ
σπέρματός σου οὐχ ὑπερόψῃ.
Break thy bread to the hungry, and lead the unsheltered poor to thy
house: if thou seest one naked, clothe him, and thou shalt not
disregard the relations of thine own seed.
(New Testament translation above is the KJV, Septuagint is from Brenton)