A large magnifying glass.
A red Bic lighter. (I’d scratch the logo off first)
MP3 player loaded with an audio version of a world history or science book, or the above mentioned shoes. Either will work.
A duck!
A working scale model of a printing press or the same of some textile machinery. I would have to ask around to know who to target 1000 yrs ago that could make sense of it, tho.
Maybe a smaller version of the time machine I’m using, set up a two way communication.
A lit stick of dynamite. Hilarity, of course!
I’d send a toy llama pulling a wagon to the Inca. Something like this. I know all the theories about why wheels weren’t put to use there but maybe an additional nudge would make an interesting change to history.
A replica of Harrison’s original marine chronometer (with the face removed to comply with the conditions of the OP). People of a thousand years ago would be able to recognize that the hands go around at a regular interval, and they would understand the implications of a clock unaffected by a ship’s movements. Trade among Europe, the western hemisphere, and east Asia would begin much earlier, before European civilization developed overwhelming technological advances.
A blue Bic lighter… but I’d leave the logo on!
Dun dun dun.
Not allowed-read the OP.
No logo with the word “bic”.
Too late to edit but, before anyone corrects me, it occurs to me that I’m a couple hundred years too early to hit the Inca. I generically (and inaccurately) used the term for the pre-Columbian people of that region.
Now I feel better 
Still an interesting idea, though. Perhaps the indigenous people will advance enough because of your donation to become a power strong enough to stop the Incas from ever taking hold?
I’d be tempted to send a tiny little printing press (minus the type), but I’m not sure they’d grok what it could be used for without them. They’d probably just make tortillas with it.
Bic is a word?
Post # 24
Maybe they would make pressed duck.
A large vibrating egg.
The OP says nothing about audio. My Ipod nano does have any writing or other text, or way to output that. If you want to complain about the logo on the back, just have a specialty one without the logo.
bad call from the officials, I say.
As far as what I would send back, my first thought was a suitcase nuke. Because some men just want to watch the world burn, I suppose.
Of course! A transtator!
First Contact my ass!
Is this the Doctor? This must be some kind of test. I am not interested in being a companion, tyvm
Oh and to answer the question. I would send a Cadbury cream egg back a thousand years. Let them figure that bit of chocolate goodness out.
I think the intent of “the officials” was pretty clear, though, so the rule is revised: No audio, either.