According to flightconnections.com, the airports that have non-stop service to both Santiago and Hong Kong (and so could be used for a connecting city) are: Sydney (SYD), Auckland (AKL), Madrid (MAD), Paris (CDG), Rome (FCO), Toronto (YYZ), London (LHR), Dallas/Fort Worth (DFW), New York (JFK), and Los Angeles (LAX).
For these countries, according to Wikipedia, a Chinese citizen of Hong Kong with a normal passport requires the following documents to enter these countries:
- France, Italy, or Spain (i.e., Schengen), the UK, or New Zealand: No visa necessary
- Canada & Australia: an electronic travel authorization (eTA)
- United States: full visa required
Note that the restrictions may be different for temporary passports, or for transit as opposed to entry; so I would not purchase any new ticket until you know precisely what kind of travel document you will have and you've triple-checked its acceptability with the airlines.
It is also possible to connect via two stops in Brazil (either São Paulo or Rio) and thence to either Johannesburg, Doha, or Dubai; if none of the above options work out, these might be worth investigating.
If your friend is issued a conventional Chinese passport as a temporary replacement, rather than a HKSAR passport, then according to the excellent answer here, she should still be able to transit in the Schengen area so long as she does not need to leave the international area of the airport. The UK, USA, Australia, New Zealand, and Canada will all require transit visas.