Fare basis MHRI almost certainly means that your flights were booked in booking class M, which is a discount economy class. Although I don't know if it's different for BKK (no obvious reason why it should be), M class is the class that my discount consolidator tickets to Oz/NZ are usually booked in on BA and QF.
Your first port of call, if you haven't already exhausted your patience with them, is to find out from the travel agent exactly what the change/cancellation/refund conditions were for this ticket. If you still have the original booking documentation (itinerary/confirmation etc.), there might be something there. Consolidator tickets can sometimes be cancelled, for a fee, after ticketing but before departure - but this will be governed as much by the contract between you and the consolidator or travel agent as by the airline's fare rules. That's why you need the documentation you got at the time of booking. Sometimes these conditions will be repeated in an itinerary or invoice given to you when you are sent the tickets. (For this purpose, don't be too put off by the "non-endorseable non-refundable" endorsement on the ticket. That relates primarily to what the airline can do directly with you on the strength of the ticket. You need to find out what the consolidator/travel agent will do for you.)
The second line is then to take bealine's excellent advice. When you discuss this with sales/res, one possibility to ask about is whether the ticket is upgradeable. This doesn't necessarily mean upgrading to a different class of service, just a different booking class which is still an economy class fare. Sometimes, upgrading to a flexible fare is possible - then what you get for the extra money is the facility to change the reservations, although almost always the original non-refundable amount remains non-refundable. If BA tells you to talk to your travel agent first about this, then you could discuss this option with them as well.
Otherwise, it will be time to ask for a favour. In many ways, you are lucky that you are dealing with BA. But there is no guarantee of anything, you may be entitled to absolutely nothing and you will pretty certainly be asking for some sort of waiver of the rules, so pitch any letter accordingly.
Good luck!
[Edited to add:]
For the official BA line on this, go to the ba.com home page. Click "contact us" at the very top of the page, which will take you to "Ask BA". Go to Answer ID 1694, which is about "Travel arrangements following the hospitalisation/death of a passenger".
Last edited by Globaliser; 2nd Oct 2003 at 03:29.