After doing about 3-5 minutes of internet search i have found what the Japanese customs for handkerchiefs are.

It is customary for men and women to keep a reusable handkerchief to wipe the sweat from their faces/hands and to dry ones hands after washing them in the bathroom. You usually take one with you wherever you go and they are also used at the end of meals to clean oneself off. You then clean it and reuse it.

What is also to note is that in Japan it is expected of the wife to make sure that their husband and children have their handkerchiefs before leaving the house so that they will be able to keep themselves presentable.

In essence she is telling Ash at the very least that, take this with you and remember me whenever you use it (presumably to keep himself clean and of course his pokemon) and more likely telling Ash "I love you" (as seen in pokemon 200 when said "I have Ash" and numerous other moments). She is basically saying don't forget that i love you even though i can't follow you where you are going so please take this with you so i can do something for you even from far away and don't forget to come back to me.

Of course seeing as this is Ash he probably takes it as "BITCHIN handkerchief!"

This ship sank a long time ago