"Ta" and "ta-ta"
Solution 1:
tata
also ta-ta, "good-bye," 1823, a word first recorded as infant's speech. Abbreviation T.T.F.N., "ta-ta for now," popularized 1941 by BBC radio program "ITMA," where it was the characteristic parting of the cockney cleaning woman character Mrs. Mopp, voiced by Dorothy Summers.
ta
1772, "natural infantile sound of gratitude" [Weekley].
At least that's what Etymology Online states.