The word καρδία in NT koine Greek, when used metaphorically, denotes the center of emotions and spiritual life, very similar to modern English. See appendix below. As such, "to open the eyes of the heart" is to see spiritual things clearly; specifically, "so that you may know the hope of His calling, the riches of His glorious inheritance in the saints" (Eph 1:18). Paul goes on at some length to enlarge this glorious prospect that awaits a true spiritual understanding of the unseen world. See V19-23
b. the center and seat of spiritual life, "the soul or mind, as it
is the fountain and seat of the thoughts, passions, desires,
appetites, affections, purposes, endeavors" (so in English heart,
inner man, etc.); α. universally: Matthew 5:8; Matthew 6:21; Mark
7:19; Luke 1:51; Luke 2:51; Luke 8:12, 15; Acts 5:3; Romans 10:9; 1
Corinthians 14:25; 2 Corinthians 6:11; Ephesians 6:5; Colossians 3:22;
1 Peter 3:4, etc.; plural: Matthew 9:4; Mark 2:6, 8; Mark 4:16 (R L
text Tr marginal reading); Luke 1:17; Luke 2:35; Luke 5:22; (Luke
24:38 R G L marginal reading; Acts 7:51 L T Tr WH text); Romans 2:15;
Romans 16:18; 2 Corinthians 3:2; Galatians 4:6; Philippians 4:7;
Ephesians 5:19 Lachmann; Hebrews 8:10 (T WH marginal reading
singular); Hebrews 10:16, etc. ἡ καρδία is distinguished from τό στόμα
or from τά χειλεα: Matthew 15:8, 18; Mark 7:6; 2 Corinthians 6:11;
Romans 10:8f; from τό πρόσωπον: 2 Corinthians 5:12; 1 Thessalonians
2:17; περιτομή καρδίας, Romans 2:29; ἀπερίτμητοι τῇ καρδία, Acts 7:51
(L T Tr WH text καρδίαις, WH marginal reading genitive καρδίας, cf.
Buttmann, 170 (148)). of things done from the heart i. e. cordially or
sincerely, truly (without simulation or pretence) the following
phrases are used: ἐκ καρδίας (Aristophanes nub. 86), Romans 6:17; and
L T Tr WH in 1 Peter 1:22, where R G ἐκ καθαρᾶς καρδίας, as in 1
Timothy 1:5; 2 Timothy 2:22; ἀπό τῶν καρδιῶν, Matthew 18:35 (ἀπό
καρδίας εὐχάριστος τοῖς θεοῖς, Antoninus 2, 3); ἐν ὅλῃ τῇ καρδία and
ἐξ ὅλης τῆς καρδίας, Matthew 22:37; Mark 12:30, 33; Luke 10:27, and
Rec. in Acts 8:37 (Deuteronomy 6:5; Deuteronomy 26:16; Psalm 118:34
()); μετ' ἀληθινῆς καρδίας, Hebrews 10:22. ἐρευναν τάς καρδίας, Romans
8:27; Revelation 2:23; δοκιμάζειν, 1 Thessalonians 2:4; γινώσκειν,
Luke 16:15 (ἐτάζειν, Jeremiah 17:10; Psalm 7:10); διανοίγειν τήν
καρδίαν (see διανοίγω, 2), Acts 16:14; ἦν ἡ καρδία καί ἡ ψυχή μία,
there was perfect unanimity, agreement of heart and soul, Acts 4:32;
τιθέναι τί ἐν τῇ καρδία (בְּלֵב and לֵב עַל שׂוּם, 1 Samuel 21:12;
Malachi 2:2; Daniel 1:8; τιθέναι ἐν στηθεσσιν, ἐν φρεσίν, etc., in
Homer), to lay a thing up in the heart to be considered more carefully
and pondered, Luke 1:66; to fix in the heart i. e. to purpose, plan,
to do something, Acts 5:4 (A. V. conceived in thy heart); also εἰς τήν
καρδίαν (L T Tr WH ἐν τήν καρδίαν) followed by the infinitive, Luke
21:14; βάλλειν εἰς τήν καρδίαν τίνος, followed by ἵνα, to put into
one's mind the design of doing a thing, John 13:2; also διδόναι
followed by an infinitive, Revelation 17:17; ἀναβαίνει ἐπί τήν καρδίαν
τίνος, followed by an infinitive, the purpose to do a thing comes into
the mind, Acts 7:23; ἐν τῇ καρδία joined to verbs of thinking,
reflecting upon, doubting, etc.: ἐνθυμεῖσθαι, διαλογίζεσθαι, Matthew
9:4; Mark 2:6, 8; Luke 3:15; Luke 5:22; λέγειν, εἰπεῖν (בְּלִבּו
אָמַר), to think, consider with oneself, Matthew 24:48; Luke 12:45;
Romans 10:6; Revelation 18:7 (Deuteronomy 8:17; Deuteronomy 9:4);
συμβάλλειν, to revolve in mind, Luke 2:19; διακρίνεσθαι, to doubt,
Mark 11:23; διαλογισμοί ἀναβαινουσι, of persons in doubt, Luke 24:38
(R G L marginal reading plural); ἀναβαίνει τί ἐπί καρδίαν, the thought
of a thing enters the mind, 1 Corinthians 2:9. β. specifically, of the
understanding, the faculty and seat of intelligence (often so in Homer
also (cf. Nägelsbach, Homer. Theol., p. 319f; Zezschwitz,
Profangräcität as above with, pp. 25f, 50); cor domicilium sapientiae,
Lactantius, de opif. dei c. 10, cf. Cicero, Tusc. 1, 9; לֵב, 1 Kings
10:2; Job 12:3; Job 17:4, etc.; (cf. Meyer on Ephesians 1:18 and
references)): Romans 1:21; 2 Corinthians 4:6; Ephesians 1:18 (Rec.
διανοίας); 2 Peter 1:19; συνιέναι τῇ καρδία, Matthew 13:15; Acts
28:27; νόειν τῇ καρδία, John 12:40. of the dullness of a mind
incapable of perceiving and understanding divine things the following
expressions occur: ἐπαχύνθη ἡ καρδία, Matthew 13:15; Acts 28:27, (from
Isaiah 6:10); πωρουν τήν καρδίαν, John 12:40; πεπωρωμένη καρδία, Mark
6:52; Mark 8:17; ἡ πώρωσις τῆς καρδίας, Mark 3:5; Ephesians 4:18;
βραδύς τῇ καρδία, slow of heart, Luke 24:25; κάλυμμα ἐπί τήν καρδίαν
κεῖται, 2 Corinthians 3:15. γ. of the will and character: ἁγνίζειν
καρδίας, James 4:8; καθαρίζειν τάς καρδίας, Acts 15:9 ῥερραντίσμενοι
τάς καρδίας, Hebrews 10:22; καρδία εὐθεῖα (cf. Winer's Grammar, 32),
Acts 8:21; πονηρά, Hebrews 3:12 (cf. 11. § 132, 24; Winer's Grammar,
194 (183)); ἀμετανόητος, Romans 2:5; γεγυμνασμενη πλεονεξίας, 2 Peter
2:14; στηρίζειν τάς καρδίας, 1 Thessalonians 3:13; βεβαιουν, in
passive, Hebrews 13:9; σκληρύνειν, Hebrews 3:8; ἡ ἐπίνοια τῆς καρδίας,
Acts 8:22; αἱ βουλαί τῶν καρδιῶν 1 Corinthians 4:5; προαιρεῖσθαι τῇ
καρδία, 2 Corinthians 9:7; κρίνειν (to determine) and ἑδραῖος ἐν τῇ
καρδία, 1 Corinthians 7:37. δ. "of the soul so far forth as it is
affected and stirred in a bad way or good, or of the soul as the seat
of the sensibilities, affections, emotions, desires, appetites,
passions": ἡ καρδία καιομένη ἦν, of the soul as greatly and peculiarly
moved, Luke 24:32; αἱ ἐπιθυμίαι τῶν καρδιῶν, Romans 1:24; στηρίζειν
τάς καρδίας, of the cultivation of constancy and endurance, James 5:8.
in reference to good-will and love: ἐήξειν τινα ἐν τῇ καρδία, to have
one in one's heart, of constant remembrance and steadfast affection,
Philippians 1:7 (`te tamen in toto pectore semper habet' Ovid. trist.
5, 4, 24); εἶναι ἐν τῇ καρδία τίνος, to be cherished in one's heart,
to be loved by one perpetually and unalterably, 2 Corinthians 7:3;
εὐδοκία τῆς καρδίας, Romans 10:1. in reference to joy and pleasure:
ηὐφράνθη ἡ καρδία, Acts 2:26 (from Psalm 15:9 ()); χαρήσεται ἡ καρδία,
John 16:22; ἀνήρ κατά τήν καρδίαν τοῦ Θεοῦ, i. e. in whom God
delights, Acts 13:22; of the pleasure given by food, Acts 14:17
((Winer's Grammar, 156 (148) note) see 2 a. above). in reference to
grief, pain, anguish, etc.: ἡ λύπη πεπλήρωκε τήν καρδίαν, John 16:6;
ὀδύνη τῇ καρδία μου, Romans 9:2; ἡ καρδία ταράσσεται, John 14:1, 27;
συνοχή καρδίας, 2 Corinthians 2:4; βάρειν τῆς καρδίας μερίμναις
βιωτικαῖς, Luke 21:34 (cf. 2 a. above); διαπρίομαι τῇ καρδία, Acts
7:54; συντετριμμένος τήν καρδίαν, Luke 4:18 R L brackets; κατενύγησαν
τῇ καρδία, Acts 2:37 (L T Tr WH τήν καρδίαν); συνθρύπτειν τήν καρδίαν,
Acts 21:13. ε. of a soul conscious of good or bad deeds (our
conscience): 1 John 3:20f (Ecclesiastes 7:22; so לֵבָב, Job 27:6; ἡ
καρδία πατασσει τινα, 1 Samuel 24:6; 2 Samuel 24:10).