I'm sorry your family is having to face this trial.
However your mother has a much better chance of recovery if she is not given blood. All kinds of major operations (including heart surgery) are performed successfully without blood and medical research indicates clearly that patients that are not given blood have shorter recovery times and have less chance of actually dying than those that do.
I am sure, as her child you are eagre to learn as much accurate medical information as possible on the subject so I will post some useful links for you to read up on.
On a psychological note, patients that suffer severe depression and trauma are much more likely to die when undergoing major operations, putting undue pressure on your mother to compromise her core beliefs (even going so far as to force her through the courts) to accept something she views as unacceptable medical treatment will be very harmful for her. Plus should she recover, your relationship will no doubt be irrevocably damaged.
It is totally normal, as her child you want the very best for your mother - but as a Witnesses she no doubt has contacted a surgeon with the necessary skills to perform the operation without blood. No surgery is without risk. There are no guarantees, even with blood there could not be a negative outcome. What your mother now needs is a skilled surgeon and a loving family. If you've found the former can you be the latter? Out of love?
Please read up on the subject and respect her wishes. She'll be fine. I don't know her, I've never met her - but she's my sister and I love her.
My prayers are with you all.
******************* ARTICLES ******************************
Heart surgery without loss of blood
http://www.dnaindia.com/report.asp?newsid=1261767
http://www.cts.usc.edu/tfct-complexheartsurgeries.html
In Depth Interview: Bloodless Transplant
http://www.wctv.tv/medicalminute/interviewheadlines/42271152.html
Bloodless surgery increasing in popularity
http://www.digitaljournal.com/article/281102
U.S Military decides to "Abstain from Blood"
http://www.youtube.com/watch?v=p5oVLbVnOqU
Noblood.Org Homepage
http://wiki.noblood.org/Main_Page#Patients_Who_Refuse_Blood_Transfusions_-_FAQs
Noblood.Org - Ask a Professional
http://www.noblood.org/ask-professional/
http://web.archive.org/web/20030402062528/wehealny.org/services/bloodless/
Bloodless surgery and medicine:
http://www.swedish.org/body.cfm?id=133
http://www.findarticles.com/p/articles/mi_m0FSL/is_1998_Nov/ai_53268419/pg_2