All pronouns common in European languages (save Basque, Hungarian, Estonian, Finnish, and Lapp) date back to Proto-Indo-European, e.g. I < **egho* 'I', me < **me* 'me', thou < **tu* 'thou', thee < **te* 'thee', etc.
These pronouns have been preserved in more descendant languages than any other word. For pronoun root *m- '1st person non-subject', I have found it in 480 out of the 482 IE languages and dialects for which I obtained data.
Pronoun roots *m- '1st person' and *t- '2nd person' are found in a cluster of language families covering northern Eurasia (Indo-European, Uralic-Yukaghir, Altaic, Gilyak, Chukchi-Kamchatkan, and Eskimo-Aleut, as well as, somewhat faintly, in Nippo-Korean). Mostly preserved in all descendants as well.
Their massive preservation in the last 5,000 to 10,000 years makes them very unlikely independent innovations in these families. Most likely they are the traces of a more ancient language, Proto-Eurasiatic. Greenberg wrote two splendid books about this linguistic phylum (Stanford University Press, 2000 and 2002).
If you are interested I wrote an essay about the ultimate origin of personal pronouns, which are very special words (changing referent with each speaker), when syntax first evolved and it began to be "unwieldy," as the first answer rightly says, to use personal names to indicate who's doing what to whom.