Will she be able to teach undergrads at Rice? If so, that will give her the equivalent of any stipends. There are probably other jobs in Shepherd, as well. At least Yale works that way- as MM students only get a minimal stipend (definitely not enough to live on!)
You may want to wait for the CCM offer to use it as levage if it is better than the Rice offer. Which ever offer is best can be used at the other school in a matter of fact way. If CCM is not better than Rice and she likes Rice I think it is fine to ask if more money is available. They will not rescind an offer bc you ask. I do think you should have an idea of how much more you want. My D did this (at a diiferent school). She gave an amt and said if they would do it she’d accept. They did. But she was confident the school was interested. There was a good amt of back and forth. There were some others schools that I think would have said no to an addl request. Still I would have told my D to ask for more just in case. If done in a straight forward, thoughtful manner I don’t think it hurts. In some cases it will be a simple no. In others you may be surprised. And this was for grad school where I think they expect more money questions from students.
A full tuition scholarship to Rice is incredible. Congratulations!
One other thought. Stipends seem to be pretty small at many grad schools. My D was bummed not to get one. But then she lucked out and got a service job at a company that is very flexible, pays well (even a 401K for part timers) and allows her to work about 16 hrs a week with good pay. During rehearsals she can reduce hours. This would have been hard in undergrad but is more manageable in grad school. I’m not sure why. I try not to be too nose-y. Less homework and papers? Fewer group projects? Better time mgmt skills? Motivation to cover her expenses (besides rent which I pay)? Probably a little of each. So just one experience… The extra money from her work was unexpected in our planning and very welcomed. One door closes, another opens.
My S might be in this same position in the next few weeks. When asking for more money, since it is music school does the student contact the Professor they will be studying with to let them know they would like to accept the offer, but need more money to make it work and ask the professor to intercede on their behalf? Or just directly ask the financial aid department? Thanks for any info on how this works.
My D started with the music financial aid office - where the initial offer came from. She wrote her request. At some point the voice faculty had to decide. We were told it could take awhile but they in the end came back quickly. I will say what she asked for was based on other offers and was a reasonable sum. But over 2 years it’s a chunk of change she could use and they could afford to give away.
Edit: the voice faculty seemed to have a set amt of scholarship dollars and I got the impression from emails (my D would send me the copies) that they were trying to guess estimate who would accept with scholarship or not. So that is why I say ask and be patient. There are dollars sloshing around and the only way to get them is to ask. You may not. But so what. At least you tried. Don’t let your ego get in the way of some dollars.
^ Thank you!
My understanding is that Rice gives full tuition for grad school but not a stipend. (Although this is just hearsay and I could be all wet.) Stipends are pretty rare for MM degrees. Musicians can generally earn more money than other types of grad students. My daughter was offered two MM spots that came with small stipend, but she chose a different school, which covered tuition but no stipends. She has been able to earn enough to cover her expenses without help from us.
Before you speak to Rice, speak the language. As set forth on their web site, your D has been awarded a full Graduate Scholarship. What you seek is a Graduate Fellowship.
http://music.rice.edu/graduate/financial.shtml
Congrats on what she has received so far!
Thank you for the great feedback everyone!
CurlyMom, did your D receive Rice acceptance via email, portal, or snail mail?
@CurlyMom I was accepted to Rice for my MM in VP also!
@Clarimom I received a phone call and an email today. My portal has not changed yet.
Oh, and also CCM. I know a little bit about both of the programs, so if you have some more specific questions, I might be able to help, even though it’s not from a current student! 
Phone call and email with the actual letter attached was how she was informed.
It really does depend upon the school and the situation. My daughter was able to negotiate her scholarship for her first year upwards substantially and also had a paying position within the department. The second year, she was the TA- Opera Administrator- for the entire department, which paid everything plus a stipend. Hours were manageable for the most part except during productions, when she found that she was working 50hour/weeks in addition to her classes and roles! Thankfully, she’s excellent at time management (one major reason she was given the job!) and the place ran like clockwork, but she slept 24 hours straight after the final set strike of the year!
Congrats on Rice and CCM, great programs! Question, has anyone heard from CIM yet?
@TheFlagmaker, are you considering for next year, or was that in the past?
Nope, I’m considering for this year! 
@TheFlagmaker PM me, my daughter currently attends CCM, so if you have any questions about there, might be able to help out.