We looked further into this and cannot replicate what you described.
When you change the subject in Helpmonks, it gets properly updated and upon a reply also sent as the "subject" in the email. This then also shows properly in the email client of the recipient.
However, in our testing, we are seeing that some email clients do not respect this change, especially when the email client is the one sending the original message. I guess this is something specific and they do that (Gmail in particular) to keep the email thread organised.
There is little we can do about this, as this is obviously client specific.
That said, have you looked at custom fields in Helpmonks? These have been specifically developed to store additional information together with the conversation, e.g. order numbers, etc. You can apply as many custom fields as you like, too.
Let me know if that helps.