Firstly, I�m so sorry for your loss and that you have this additional stress in what may already be an really upsetting time.
The answer to your question (regardless of the debate that has gone before) actually hinges on whether or not a Will exists. If it does it will name an Executor and one or more Beneficiaries. Arranging the funeral is a responsibility of the Executor (although can be delegated). The Exec is not necessarily a Beneficiary. A Will will usually say what the deceased would like done with their body but it�s a little known fact that the Exec is free to disregard this instruction. Unless there are specific words for the headstone in the Will (this is unlikely), the Exec chooses.
If no Will exists, you�re in a different ballgame and incidentally one where your son does have more rights. If intestate, there is a strict order of who the law recognises. The girlfriend will not appear on this list. There�s some good stuff here:
http://www.direct.gov.uk/en/RightsAndResponsib ilities/Death/index.htm
On the practical side, most headstones don�t go up for 6 months (the ground is still settling) so you have time to resolve this amicably.
I really hope that you�re able to do this. In the meantime, the CAB might be a good port of call.
Good luck.