A common technique is to put an identifier in the Subject "[Contact:foo]" like you say, with a note in the body telling the customer not to change the Subject so the email will be routed correctly.
"Please reply to this email without changing the subject, so your matter can be promptly attended to".
Obviously you'll have to ignore things like "Re:", "Fwd:" etc when parsing inbound subjects.
This is safer than putting something in the body and requiring the customer to include the original content in his/her reply.