This is not specifically Russian.
Here's a quote from Wikipedia:
From the late 18th century white handkerchiefs were waved, generally
by women (men usually waved their hats), to demonstrate approval at
public events such as processions or political rallies
This has nothing to do with neither headdress (which also called
платок in Russian) nor with Russian traditions apart from fashion among westernized circles of Russian aristocracy. Even more specifically - Russian aristocracy was under a heavy French influence - and you can a lot of references for waving a handkerchief in the French literature of the XIX century.
Back to the linguistic part of the question - there's no symbolic meaning of any kind, it's just that in modern Russian "махать платочком" has an ironical (yet this irony is of a kind nature) connotation.
Ты вот уедешь на поезде, а я буду стоять на пероне и махать платочком. И вся рыдать.