Thayer (from 1889) seems to have gotten mixed up with a cognate, μονόω G3443 which is used of widows "bereaved" as in 1 Tim 5:5. μόνος is never used that way, per BDAG. And even Thayer lists John 17:3 under "only" as does BDAG. This is the adjective in question:
μόνος, η, ον (Pind.+ [as μοῦνος as early as Hom.]) ① pert. to being
the only entity in a class, only, alone adj. ⓐ with focus on being the
only one α. used w. verbs like εἶναι, εὑρίσκεσθαι, καταλείπειν: μόνος
ἦν ἐκεῖ Mt 14:23; cp. J 8:16. Λουκᾶς ἐστιν μόνος μετʼ ἐμοῦ 2 Ti 4:11.
εὑρέθη Ἰησοῦς μόνος Lk 9:36. μόνην με κατέλειπεν 10:40 (w. inf.
foll.); pass. κατελείφθη μόνος J 8:9; cp. 1 Th 3:1. κἀγὼ ὑπελείφθην
μόνος I am the only one left (Theseus Hist. [Roman times]: 453 Fgm. 2
Jac. μόνος περιλειφθείς of the only survivor of a battle) Ro 11:3 (cp.
3 Km 19:10, 14; Job 1:15 al.).—Ac 15:33 [34] v.l. β. used w. a noun
(TestJob 10:1 τοῖς ξένοις μόνοις; AssMos Fgm. d p. 64 λόγῳ μόνῳ) τὰ
ὀθόνια μόνα Lk 24:12. μόνοι οἱ μαθηταὶ ἀπῆλθον J 6:22. μόνος ὁ
ἀρχιερεύς Hb 9:7.—Cp. Mt 12:4 (Jos., Ant. 15, 419 τ. ἱερεῦσιν ἐξὸν ἦν
μόνοις). οὗτοι μόνοι συνεργοί Col 4:11. μόνῳ πνεύματι AcPl Ant 13, 18
(μόνον πνεύματι Aa I 237, 3). Papias (3:2) αὐτὸν μόνον τῆς κεφαλῆς
ὄγκον.—Used w. pronouns (μόνος αὐτός: Nicol. Dam.: 90 Fgm. 130, 23 p.
407, 21 Jac.; Ps.-Demetr., De Eloc. 97; 2 Macc 7:37; Philo, Agr. 39;
Jos., Ant. 8, 405, C. Ap. 1, 49); αὐτῷ μόνῳ λατρεύσεις (Dt 6:13 v.l.;
cp. Jos., Ant. 3, 91 τοῦτον μ.) Mt 4:10; Lk 4:8.—Mt 18:15; Mk 6:47;
9:2; J 6:15. εἰς ἑαυτὸν μόνον Gal 6:4. σὺ μόνος … ; (1 Km 21:2;
TestZeb 4:12; cp. σὺ μόνος ApcSed 15:1) are you the only one? (Field,
Notes 82) Lk 24:18; ἐγὼ μ. (En 6:3; TestJud 3:1; ApcMos 27) Ac 26:14
v.l.; 1 Cor 9:6; GJs 1:3; ὑμεῖς μόνοι 1 Cor 14:36 (cp. Just., D. 19,
2). γ. w. a negative and w. ἀλλά foll.: οὐκ ἐπʼ ἄρτῳ μόνῳ … , ἀλλʼ …
(Dt 8:3) Mt 4:4=Lk 4:4. οὐ μόνον τὸ σῶμα … ἀλλὰ πολλοστόν AcPlCor 2:27
οὐκ ἐγὼ μ. … , ἀλλὰ καί … Ro 16:4; 2J 1. οὐκ ἐγράφη δὲ διʼ αὐτὸν
μόνον, ἀλλὰ καί Ro 4:23. οὐκ αὐτὸν δὲ μόνον, ἀλλὰ καί Phil 2:27.
Pleonast. w. εἰ μή after a neg. not … except … alone (Lydus, Magist.
1, 18 p. 22, 22) Mt 12:4; 17:8; 24:36; Mk 9:8 v.l. (for ἀλλὰ … μόνον);
Lk 5:21; 6:4; Phil 4:15; Rv 9:4 v.l. δ. μόνος θεός (cp. Simonides,
Fgm. 4, 7 Diehl θεὸς μόνος; Da 3:45; SibOr 3, 629; PGM 13, 983) the
only God 1 Ti 1:17; Jd 25 (GDelling, TLZ 77, ’52, 469–76). W. article
preceding ὁ μόνος θ. (EpArist 139; Philo, Fuga 71; Just., D. 126, 2
τοῦ μόνου καὶ ἀγεννήτου θεοῦ υἱόν; ὁ θεὸς μόνος 4 Km 19:15, 19; Ps
85:10; Is 37:20. Cp. ENorden, Agn. Theos 1913, 245, 1) J 5:44 (without
θεοῦ v.l.). ὁ μ. ἀληθινὸς θεός the only true God 17:3 (Demochares
[c. 300 B.C.]: 75 Fgm. 2 Jac. τὸν Δημήτριον οἱ Ἀθηναῖοι ἐδέχοντο …
ἐπᾴδοντες ὡς εἴη μόνος θεὸς ἀληθινός, οἱ δʼ ἄλλοι καθεύδουσιν ἢ
ἀποδημοῦσιν ἢ οὐκ εἰσίν. γεγονὼς δʼ εἴη ἐκ Ποσειδῶνος καὶ Ἀφροδίτης
‘the Athenians welcomed Demetrius … adulating him with the surmise
that he was the only real god, whereas p 659 others were
slumbering, or taking a trip, or simply did not exist; and that he was
probably a descendant of Poseidon and Aphrodite’). τὸ ὄνομα τ.
ἀληθινοῦ καὶ μόνου κυρίου 1 Cl 43:6 (cp. Just., D. 55, 2 κύριος μ.).
μ. σοφὸς θεός the only wise God Ro 16:27 (Philo, Fuga 47 ὁ μ. σοφός;
Heraclitus, Fgm. 32 ἓν τὸ σοφὸν μοῦνον). ὁ μ. δεσπότης the only one
who is master Jd 4 (cp. Jos., Bell. 7, 323; 410). ὁ μακάριος καὶ μόνος
δυνάστης 1 Ti 6:15.—Vs. 16; Rv 15:4. ⓑ with focus on being helplessly
alone: alone, deserted, helpless (Hom. et al.; BGU 180, 23 [172 A.D.]
ἄνθρωπος πρεσβύτης καὶ μόνος τυγχάνων; 385, 4; Wsd 10:1; TestJos 1:6;
La 1:1) οὐκ ἀφῆκέν με μόνον J 8:29; 16:32ab (ἀφ. μόν. as Dio Chrys. 46
[63], 2). ⓒ with focus on isolation: isolated, by itself (cp. Bar
4:16; En 28:1; TestJud 5:3; JosAs 2:16; Ar. 11:2) ἐὰν μὴ ὁ κόκκος τ.
σίτου … ἀποθάνῃ, αὐτὸς μόνος μένει J 12:24. In Hv 3, 9, 2 μ. refers to
selfish Christians who isolate themselves fr. the needs of the hungry.
② a marker of limitation, only, alone, the neut. μόνον being used as
an adv. (Aeschyl., Hdt. et al.) ⓐ limiting the action or state to the
one designated by the verb (TestAbr A 4 p. 81, 25 [Stone p. 10];
TestJos 16:4; JosAs 24:10) Mt 9:21; 14:36; Mk 5:36; Lk 8:50; 1 Cor
7:39; 15:19; Gal 1:23; Phil 1:27; 2 Th 2:7; Hv 3, 2, 1.—οὐκ ἀλλὰ μ. Mt
8:8. ⓑ w. a noun (Just., A I, 22, 1 μ. ἄνθρωπος) or pron., to separate
one pers. or thing fr. others: Mt 5:47; 10:42; Ac 18:25; Ro 3:29; Gal
2:10; Hb 9:10; Hm 12, 4, 7; τοῦτο μ. Gal 3:2. ⓒ used w. negatives α.
μ. μή only not, not only (POxy 2153, 22; TestJob 45:1) Gal 5:13. οὐ
(μὴ) μ. 4:18; Js 1:22; 2:24 (s. β below). οὐ μ. … ἀλλά (without καί
when the second member includes the first. X., Cyr. 1, 6, 16; Diod S
4, 15, 1; Dio Chrys. 1, 22; 62; 64 [14], 7; Just., A I, 2, 1; B-D-F
§448, 1) Ac 19:26 (but some mss. add καί); 1 Th 1:8; 1J 5:6. οὐ (or
μὴ) μ. … , ἀλλὰ καί not only … , but also (PMich 209, 12 [c. 200
A.D.]; TestJos 10:3; Jos., Bell. 3, 102; Just., A I, 5, 4 al.) Mt
21:21; J 5:18; Ac 21:13; 26:29; 27:10; Ro 1:32; 9:24; 13:5; 2 Cor
8:10, 21; 9:12; Eph 1:21; Phil 1:29; 1 Th 2:8; 2 Ti 2:20; Hb 12:26; 1
Pt 2:18; Qua. οὐ (μὴ) … μ., ἀλλὰ καί J 11:52; 12:9; 13:9; 17:20; Ro
4:12, 16; Phil 2:27; 1 Th 1:5 al. οὐ … μόνον ἀλλὰ καί 1J 2:2. οὐδέπω …
, μ. δέ not yet … , but … only Ac 8:16. οὐ μ. δέ, ἀλλὰ καί not only
this, but also (ellipsis w. supplementation of what immediately
precedes; Mitt-Wilck. II/2, 26, 9=27, 9 [108 B.C.]; cp. Sb 7616 [II
A.D.]; Wsd 19:15; TestJob 35:1; Just., A I, 49, 5; B-D-F §479, 1; s.
Rob. 1201ff) Ro 5:3, 11; 8:23; 9:10; 2 Cor 8:19. οὐ μόνον δὲ … ἀλλὰ
καί (TestZeb 3:7) Ac 19:27; 2 Cor 7:7; 1 Ti 5:13. μὴ μ., ἀλλὰ πολλῷ
μᾶλλον not only, … but much more Phil 2:12. οὐδὲν (μηδὲν) … εἰ μὴ …
μόνον (TestAbr B 11 p. 116, 3 [Stone p. 80]; TestJob 11:7; Ar 13, 7)
Mt 21:9; Mk 6:8. μηδενὶ … εἰ μὴ μ. Ac 11:19. On 1–2c s. KBeyer,
Semitische Syntax im NT ’62, 126–29. β. in isolation οὐκ ἐκ πίστεως
μόνον not by faith viewed in isolation Js 2:24 (NEB: ‘not by faith in
itself’; Goodsp.: ‘not simply by having faith’; sim. Moffatt; s.
Athanasius Alexandrinus, De Virginitate PGM 28, 260c; cp. Clem.,
Strom. 3, 15 οὐ γὰρ μόνον ἡ εὐνουχία δικαιοῖ=being a eunuch does not
of itself justify. S. also πίστις 2dδ.) ⓓ ἵνα μόνον solely in order
that 12:8; μόνον ἵνα Gal 6:12. ③ κατὰ μόνας (Thu. 1, 32, 5; X., Mem.
3, 7, 4; Menand., Epitr. 988 S. [658 Kö.], Fgm. 146 Kö.; Polyb. 4, 15,
11; Diod S 4, 51, 16; Gen 32:16; Ps 4:9; Jer 15:17; 1 Macc 12:36;
TestJos 4:1; Jos., Vi. 326, Ant. 17, 336 al.—Also written καταμόνας;
cp. BGU 813, 15 in APF 2, 1903, 97) alone γίνεσθαι κ. μ. be alone
(Syntipas p. 9, 16) Mk 4:10.—Lk 9:18; Hm 11:8.—B-D-F §241, 6.—B. 937.
DELG. Schmidt, Syn. IV 535–39. M-M. EDNT. Sv.
Arndt, W., Danker, F. W., & Bauer, W. (2000). A Greek-English lexicon
of the New Testament and other early Christian literature (3rd ed.,
pp. 658–659). Chicago: University of Chicago Press.
The "true God" of 1-John 5 is also the Father.