The word ἱστορίης that you see in the text of Herodotus is actually in the genitive case, literally "of research / of investigation." So Herodotus introduces his work as an "exposition of research" (ἱστορίης ἀπόδεξις). The word that we render as "history" is (in the nominative case) ἱστορίη in Herodotus' Ionic dialect, or ἱστορία in standard Greek. Cf. Liddell & Scott's comprehensive Greek Lexicon:
ἱστορ-ία, Ion. -ιη, ἡ,
inquiry, ἱστορίῃσι εἰδέναι τι παρά τινος Hdt.2.118, cf. 119; ἡ περὶ φύσεως ἱ. Pl.Phd.96a; αἱ περὶ τῶν ζῴων ἱ. Arist.Resp.477a7,al.; ἡ ἱ. ἡ περὶ τὰ ζῷα Id.PA674b16; ἡ ζωικὴ ἱ. ib.668b30; περὶ φυτῶν ἱ., title of work by Theophrastus; systematic or scientific observation, Epicur.Ep.1p.29U.: abs., of science generally, ὄλβιος ὅστις τῆς ἱ. ἔσχε μάθησιν E.Fr.910 (anap.); of geometry, Pythag. ap. lamb.VP18.89: in empirical medicine, body of recorded cases, Gal.1.144; mythology, Ἡσίοδον πάσης ἤρανον ἱστορίης Hermesian.7.22.
knowledge so obtained, information, Hdt.1 Praef., Hp.VM20; ὄψις ἐμὴ καὶ γνώμη καὶ ἱ. Hdt.2.99; πρὸς ἱστορίαν τῶν κοινῶν for the knowledge of . . , D.18.144; ἡ τῆς ψυχῆς ἱ. Arist.de An.402a4.
written account of one's inquiries, narrative, history, prob. in this sense in Hdt.7.96; αἱ τῶν περὶ τὰς πράξεις γραφόντων ἱ. Arist.Rh.1360a37, Po.1451b3, Plb.1.57.5, al.; ἐκ τῶν ἱστοριῶν καὶ ἐκ τῶν ἄλλων μαρτυριῶν OGI13.12 (iii B.C.); αἱ Μαιανδρίου ἱ. Inscr.Prien.37.105; κοινὴ ἱ. general history, D.H.1.2; ἱ. Ἑλληνική, Ῥωμαϊκή, Plu.2.119d; restricted by some to contemporary history, Lat. rerum cognitio praesentium, Verr.Flacc. ap. Gell.5.18: generally, story, account, Call.Aet.3.1.7.
If you want to transliterate the word using Roman letters, you have the choice of either transliterating from Herodotus' own dialect (historiē) or from standard Greek (historiā).