I know that they, them, and their did not exist in Old English. What language are they derived from?
Old English had a set of plural pronouns that were very similar to the masculine/feminine pronouns, differing only in the vowels. The third person plural pronoun was:
Nom: hīe [hiːə], Acc: hīe, Dat: him, Gen: hira
These gradually fell out of use to be replaced by the Old Norse word þeir, originally meaning "those". This was partly because the sound changes from Old English to Middle English would have caused many of the 3rd-person pronouns to become identical. In particular, if the word hīe had not been replaced by þeir, it would eventually have been pronounced identically to "she"!