I hate to dump a lengthy quote here, but I have a great scholarly resource available on the meaning of this word that gives TONS of extrabiblical quotations. Here are a bunch of places it occurs in other literature:
συνείδησις, εως, ἡ (συνεῖδον)
① awareness of information about someth., consciousness (Democr.,
Fgm. 297 σ. τῆς κακοπραγμοσύνης; Chrysipp. in Diog. L. 7, 85 τὴν
ταύτης συνείδησιν; Eccl 10:20; Sir 42:18 v.l.; Jos., Ant. 16, 212;
Just.; Theoph. Ant. 2, 4 [p. 102, 8]) w. obj. gen. συνείδησις ἁμαρτιῶν
consciousness of sin Hb 10:2 (Diod S 4, 65, 7 διὰ τὴν συνείδησιν τοῦ
μύσους; Philo, Det. Pot. Ins. 146 οἱ συνειδήσει τῶν οἰκείων ἀδικημάτων
ἐλεγχόμενοι, Virt. 124 σ. ἁμαρτημάτων). συνείδησις θεοῦ
consciousness, awareness of God 1 Pt 2:19 (s. ESelwyn, 1 Pt ’46,
176–78). Opp. σ. τοῦ εἰδώλου in awareness that this is an idol 1 Cor
8:7a v.l. (for συνηθείᾳ).
② the inward faculty of distinguishing right and wrong, moral
consciousness, conscience (Menand., Monost. 597 ἅπασιν ἡμῖν ἡ
συνείδησις θεός comes close to this mng.; cp. 654; Dionys. Hal., De
Thuc. 8 μιαίνειν τὴν ἑαυτοῦ συνείδησιν; Heraclit. Sto., 37 p. 54, 8
σ. ἁμαρτόντος ἀνθρώπου; Ps.-Lucian, Amor. 49 οὐδεμιᾶς ἀπρεποῦς
συνειδήσεως παροικούσης; Hierocles 14, 451; Stob., Flor. 3, 24 [I
601ff H.] quotes sayings of Bias and Periander on ὀρθὴ or ἀγαθὴ
συνείδησις; PRyl 116, 9 [II A.D.] θλιβομένη τῇ συνειδήσει περὶ ὧν
ἐνοσφίσατο; Mitt-Wilck. II/2, 88 I, 35 [II A.D.]; BGU 1024 III, 7;
PFlor 338, 17 [III A.D.] συνειδήσει=‘conscientiously’, also s. 3,
below; Wsd 17:10; Jos., Ant. 16, 103 κατὰ συνείδησιν ἀτοπωτέραν;
TestReub 4:3; TestJud 20:2 v.l.; συνείδησιν μολύνειν Hippol., Ref. 9,
23, 4) w. subj. gen. Ro 2:15; 9:1; 1 Cor 10:29a; 2 Cor 1:12; 4:2;
5:11; Hb 9:14 al.; ἡ ἰδία σ. 1 Ti 4:2. Opp. ἄλλη σ. another’s
scruples 1 Cor 10:29b; διὰ τὴν σ. for conscience’ sake (cp. OGI 484,
37 διὰ τὸ συνειδός; Ps.-Dio Chrys. 20 [37], 35) Ro 13:5; 1 Cor 10:25,
27f; τὸ μαρτύριον τῆς σ. 2 Cor 1:12, cp. σ. as the subj. of
μαρτυρεῖν Ro 9:1; cp. 2:15, or of ἐλέγχειν J 8:9 v.l. (s. ἐλέγχω 2).
W. attributes: σ. ἀγαθή a good conscience (cp. Herodian 6, 3, 4;
PRein s.v. καλός 2b) Ac 23:1; 1 Ti 1:5; 1 Pt 3:21 (on the topic cp.
FSokolowski, Lois sacrées des cités grecques, Supplément ’62 no. 108,
4–7 ‘one who enters the temple … must be pure, not through bathing but
in mind’); ἔχειν ἀγαθὴν σ. (cp. ἐλευθέραν ἐχ. τὴν σ. Did., Gen. 89,
11) 1 Ti 1:19; 1 Pt 3:16. Also ἐν ἀγαθῇ σ. ὑπάρχειν 1 Cl 41:1. ἐν
ἀμώμῳ καὶ ἁγνῇ συνειδήσει περιπατεῖν Pol 5:3 (μετὰ συνειδήσεως ἀγαθῆς
τελευτᾶν Hippol., Ref. 9, 26, 4); cp. 1 Cl 1:3. σ. ἀσθενής a weak
conscience, indecisive because of being bound to old ways 1 Cor 8:7;
cp. vss. 10, 12. σ. ἀπρόσκοπος Ac 24:16; καθαρὰ σ. 1 Ti 3:9; 2 Ti
1:3; 1 Cl 45:7; καθαρὸς τῇ σ. ITr 7:2; καλὴ σ. Hb 13:18; 2 Cl 16:4.
σ. πονηρά a bad conscience or a consciousness of guilt (s. καρδία
1bδ) Hb 10:22; D 4:14; B 19:12; Hm 3:4. ἡ σ. μολύνεται 1 Cor 8:7.
μιαίνεται Tit 1:15 (s. Dionys. Hal. above). καθαριεῖ τ. συνείδησιν
ἡμῶν ἀπὸ νεκρῶν ἔργων Hb 9:14. κατὰ συνείδησιν (s. on this Vett. Val.
210, 1) τελειῶσαί τινα vs. 9.
③ attentiveness to obligation, conscientiousness (for ins s. New
Docs 3, 85; pap.) μετὰ συνειδήσεως conscientiously 1 Cl 2:4; ἐν
ὁμονοίᾳ συναχθέντες τῇ σ. assembled in concord, with full
consciousness of our duty 1 Cl 34:7.—MKähler, Das Gewissen I 1, 1878,
RE VI 1899, 646ff; RSteinmetz, Das Gewissen bei Pls 1911; MPohlenz,
GGA 1913, 642ff, Die Stoa ’48; ’49 (index), ZNW 42, ’49, 77–79;
HBöhlig, Das Gewissen bei Seneka u. Pls: StKr 87, 1914, 1–24;
FTillmam, Zur Geschichte des Begriffs ‘Gewissen’ bis zu den paulin.
Briefen: SMerkle Festschr. 1922, 336–47; FZucker,
Syneidesis-Conscientia 1928; TSchneider, D. paulin. Begriff d.
Gewissens (Syneidesis): Bonner Zeitschr. f. Theol. u. Seelsorge 6,
1929, 193–211, D. Quellen d. paul. Gewissensbegr.: ibid. 7, 1930,
97–112; BSnell, Gnomon 6, 1930, 21ff; MDibelius Hdb.2 ’31 exc. on 1 Ti
1:5; HOsborne, Σύνεσις and σ.: ClR 45, ’31, 8–10, Συνείδησις: JTS 32,
’31, 167–79; GRudberg, JAEklund Festschr. ’33, 165ff; GJung,
Συνείδησις, Conscientia, Bewusstsein: Archiv f. d. gesamte Psychologie
89, ’34, 525–40; WAalders, Het Geweten, ’35; CSpicq, La conscience
dans le NT: RB 47, ’38, 50–80; BReicke, The Disobedient Spirits and
Christian Baptism ’46, 174–82; JDupont, Studia Hellenistica 5, ’48,
119–53; HClavier, Συν., une pierre de touche de l’Hellénisme
paulinien, announced in Studia Paulina [JdeZwaan Festschr.] ’53, p. 80
n. 1; CPierce, Conscience in the NT, ’55; BReicke, TZ 12, ’56, 157–61,
esp. 159; DMariella Jr., The NT Concept of Conscience, diss.
Vanderbilt ’59; PDelhaye, Studia Montis Regii (Montreal) 4, ’61,
229–51; JStelzenberger, Syneidesis im NT, ’61; MThrall, NTS 14,
’67/68, 118–25; BHarris, Westminster Theol. Journal 24, ’62, 173–86;
RJewett, Paul’s Anthropological Terms, ’71, 402–46; HEckstein, Der
Begriff Syneidesis bei Paulus ’83; GSelby, The Meaning and Function of
σ. in Hb 9 and 10: Restoration Qtrly 28, ’86, 145–54 (internal
awareness of sin); PGooch, Conscience in 1 Cor 8 and 10: NTS 33, ’87,
244–54; PTomson, Paul and the Jewish Law (CRINT III/1) ’90, 208–20
(‘consciousness’); EBorgh, La notion de conscience dans le NT:
Filología Neotestamentaria 10, 1997, 85–98.—RAC X 1025–1107; BHHW I
564f.—New Docs 3 no. 69. DELG s.v. οἶδα C. M-M. EDNT. TW. Spicq. Sv.1
Since συνείδησις appears in the LXX and numerous other works, it's safe to say that it is not exclusively Pauline. According to Philip Bosman, the word comes from Greek philosophy and was often used by writers who attempted to integrate Greek (Hellenistic) and Jewish philosophy.2 We can see this heavily in the writings of Philo, who was a contemporary of St. Paul (although a couple decades older). St. Paul had a similar mission as he was evangelizing to Greek Gentiles. I believe St. Paul would have come across Philo's writings, but it is also possible that both drew from a common source that is today unknown, or that they simply were both influenced by Stoic philosophy as well as the philosophy of Plato.
One thing is for sure: the Stoic philosophers used the term συνείδησις often.3 Responding to whether or not St. Paul commingled early Christian teaching with Platonism (or other Hellenistic philosophy), Basil Tsakonas writes,
Where has Paul drawn the term 'awareness' from? Assuredly from the
Hellenistic environment... not only the term 'awareness' but also many
other terms that were in use during those years. However he
transforms the content of those terms to such an extent, that they are
rendered purely Christian-centered and they have discarded their
philosophical and specifically Stoic character. The terminology is
common, but that which renders it different is the new spirit which is
given to those terms under the influence of the new religion.3
Just as St. John often did (such as with λόγος), St. Paul used concepts from Greek philosophy but used them in a distinctively Christian manner. This pattern was followed by most of the early Church fathers.
For an exhaustive comparison of Philo and Paul's use of συνείδησις, as well as its use throughout history and philosophy, cf. 2 and 3.
Sources:
1 William Arndt, Frederick W. Danker and Walter Bauer, A Greek-English Lexicon of the New Testament and Other Early Christian Literature, 3rd ed. (Chicago: University of Chicago Press, 2000), 967-68.
2 Philip Bosman. Conscience in Philo & Paul: A Conceptual History of the Synoida Word Group. (Tübingen, Germany: Coronet Books, 2003).
3 Basil G. Tsakonas. The Teaching on Conscience by Apostle Paul. (Athens, Greece: Library of Athens Educative Society #51, 1968), 68-9.