Without reading any of the comments, here is my 2c!
My great-grandparents opened many accounts for me at various banks when I was very young, as well as purchased CD's. Why so many? They were still reeling from the depression era and didn't trust enough to put all of their eggs in one basket, so to speak. All of them were local banks and anyone could deposit money, so that was never a big deal- they didn't have to be present or anything. (I'm pretty sure you can deposit as long as you have the acct # to any acct, you just can't withdraw or make changes.)
The CD's were the ones that gained me the most interest and couldn't be cashed in until I was an "adult" at 18/19 yrs old. These are your best investment if you're looking to put money in long-term.
It's not going to do any harm to have multiple accounts for her. You could even set the one from Grandma up as her "college fund" or "car fund" and the one for her as something you will match portions of, etc and she can withdraw from occasionally, etc. So that each acct has different rules for her, just like adults do!
The only snafu with mine was that my great-grandparents died long before any of my CD's matured or my parents allowed me to withdraw from the savings, so it was a little bit of a pain in the butt to have to go to the bank with the executor of their estate, documents of their death certificates, etc. and close out the accounts. It was worth it though- it really made me appreciate (even as a spoiled brat of 18 yrs old!) that someone cared enough to put money aside for my future.
One thing I might suggest is that you don't tell her the amounts that are in bigger accounts. I spent many days dreaming of how I would blow the cash already there instead of saving my own money and thinking about my goals realistically. I also wish I had not had access to the money until, say, 25. I really wasn't mature enough at 18 to handle access to the accounts and, on a personal note, ended up in a bad marriage at 21 where he found out about the biggest (yet untouched) account and spent the entire amount (which was to be for college) behind my back, and got away with it since we were legally wed. Of course this could have happened to me at any age, but I think if I had had a few more years to get my life together before having that access, it would have been less likely. Just some thoughts!