I don’t think anyone has the answer. The OED doesn’t. Here’s their entry:
[Origin unascertained.
(In Wallis’s Room for the Cobbler of Gloucester (1668) 4 tantrum appears as a Welshman’s mispronunciation of anthem, but apparently has no connexion with this word.)]
An outburst or display of petulance or ill-temper; a fit of passion. Freq. in pl. Now often spec. a fit of bad temper in a young child.
1714 E. VERNEY Let. 30 Oct. in M. M. Verney Eighteenth Cent. Memoirs (1930) II. xxi. 18 Our lady has had some of her tanterums as Vapors comeing out etc. 1748 FOOTE Knights II. Wks. 1799 I. 84 None of your fleers!..Your tantrums!You are grown too headstrong and robust for me. 1754 J. SHEBBEARE Matrimony (1766) I. 122 Where did the Wench get these Tantarums into her Head? 1776 MRS. DELANY in Life & Corr. Ser. II. (1862) II. 206 Treating him with some contempt when he is in his tantrums. 1824 W. IRVING T. Trav. I. 217 An author, who was always in a tantrum if interrupted. 1837 DISRAELI Venetia I. vi, He goes into his tantarums at the abbey. 1884 Times 12 Mar. 3 The defendant told him not to get into a tantrum. 1927 A. CONAN DOYLE Case-Bk. Sherlock Holmes xi. 283 Sir Robert was in one of his tantrums. 1966 M. FRAYN Russian Interpreter xxii. 103 %u2018Let’s put all these books away in the case again,%u2019 she said coaxingly, as if Proctor-Gould had thrown his toys about in a tantrum. 1979 N. GORDIMER Burger’s Daughter 329 If we’d still been children, I might have been throwing stones at him in a tantrum.