The word παῖς can cover the terms child, son/daughter, and servant/slave in ancient Greek. But I can't imagine a case where you'd confuse a servant into thinking you were addressing him as "son," or your son into thinking you were addressing him as "servant," any more than you'd confuse either of those persons by calling them "boy" in English. So that you can study the full range of meanings for παῖς, I've reprinted below the full definition from Liddell & Scott's authoritative Greek-English Lexicon :
παῖς, also παῦς (q. v.), παιδός, ὁ, ἡ, gen. pl. παίδων, Dor. παιδῶν Greg.Cor.p.317 S.; dat. pl. παισί, Ep.
παίδεσσι Od.3.381, etc.; in early Ep. freq. disyll. in nom. πάϊς, e. g. when forming part of two different feet, Il.2.609, 5.704, etc.; prob. also in the fifth foot, 9.57, 11.389; and before bucolic diaeresis, 2.205, al.; also in Lyr., Sapph.38, 85; and in Boeot., IG7.690, al. (Tanagra), cf. πῆς; πάϊ [ᾰῑ] Od.24.192 (παιδ- is never disyll. in oblique cases in Hom.); acc. πάϊν A. R.4.697, AP3.8 (Inscr. Cyzic.), 9.125; gen. παϊδός Epigr. ap. Luc.Symp.41; dat. παϊδί prob. in Anacr.17:
in relation to Descent, child, whether son, Il. 2.205,609, al. (with special reference to the father, opp. τέκνον, q.v.): pl., Th.1.4, etc.; or daughter, Il.1.20,443, 3.175; παῖδες ἄρρενες καὶ θήλειαι Pl.Lg.788a; παῖς, opp. κόρα, Berl.Sitzb.1927.7 (Locr., v. B.C.); of an adopted son, ἀλλά σε παῖδα ποιεύμην Il.9.494; παίδων παῖδες, τοί κεν μετόπισθε γένωνται 20.308, cf. Pi.N.7.100, Inscr.Cypr.135.11 H., etc.; Ἀγήνορος παῖδες ἐκ παίδων E.Ph.281; freq. in orators of legal issue, Isoc.19.9, Is.7.31, etc.; of animals, A.Ag.50 (anap.).
metaph., ἀμπέλου π., of wine, Pi.N.9.52; χορῶν ἐραστὴς κισσὸς ἐνιαυτοῦ δὲ παῖς Chaerem.5; ὀρείας πέτρας π., of Echo, E.Hec.1110; ὅρκου π. ἀνώνυμος, of the penalty of perjury, Orac. ap. Hdt.6.86.γ; ἄναυδοι π. τᾶς ἀμιάντου, of fishes, A.Pers.578 (lyr.).
periphr., οἱ Λυδῶν παῖδες sons of the Lydians, i. e. the Lydians, Hdt.1.27, cf. 5.49; π. Ἑλλήνων A.Pers.402; οἱ [Ἀσκληπιοῦ] π., i. e. physicians, Pl.R.407e; οἱ ζωγράφων π. painters, Id.Lg.769b; παῖδες ῥητόρων orators, Luc. Anach.19; π. ἰατρῶν, π. πλαστῶν καὶ γραφέων, Id.Dips.5, Im.9; cf. υἱός 2.
in relation to Age, child, boy or girl, νέος π. Od.4.665; παῖδες νεαροί Il.2.289; σμίκρα π. Sapph.34: with another Subst., π. συφορβός boy-swineherd, Il.21.282; παῖδα κόρην γαμεῖν Ar.Lys.595; ἐν παισὶ νέοισι π. Pi.N.3.72; π. ἔτ' ὤν A.Ch. 755, cf. Il.11.710; ἔτι π. Pl.Prt.310e; παιδὸς μηδὲν βελτίων ib.342e: distd. from παιδίον, μειράκιον, Hp.Hebd.5, cf. X.Smp.4.17, Cyr.8.7.6, 1.2.4; ἐκ παιδός from a child, Pl.R.374c; ἐκ παιδὸς εἰς γῆρας Aeschin.1.180; ἐκ τῶν παίδων εὐθύς Pl.Lg.694d, cf. R.386a; ἀκούων τῶν παίδων εὐθύς Id.Lg.642b; εὐθὺς ἐκ παίδων ἐξελθών D.21.154; ἡλικίαν ἔχειν τὴν ἄρτι ἐκ π. to be just out of one's childhood, X.HG5.4.25; ἐκ μικρῶν π. Arist.Pol. 1336a14; [Ἡρακλῆς] ἐν παισὶν ὄφεις ἀπέκτεινεν D.C.56.36; ἐν παισὶ (v.l. παιδὶ) ποιμαίνων Hdn.6.8.1; χορηγεῖν παισί (cf. χορηγέω 11): prov., τοῦτο κἂν π. γνοίη Pl.Euthd.279d; δῆλον τοῦτό γε ἤδη καὶ παιδί Id.Smp.204b; παῖδας [τοὺς πρὸ αὐτοῦ] ἀπέφηνε Luc.Peregr.11, cf. Alex.4; ἔνι τις καὶ ἐν ἡμῖν π., of the superstitious fears of a child, Pl. Phd.77e, cf. Porph.Abst.1.41.
in relation to Condition, slave, servant, man or maid (of all ages), παῖ, παῖ A.Ch.653, cf. Ar. Ach.395, Epicr.5.2, etc.; παῖ, παιδίον Ar.Nu.132: pl., of the crew of a ship, D.33.8. (From Παϝις, cf. παῦρος, Lat. puer.)