N.,
I can see your concern about setting up a bad precendent that might allow your daughter to think that there is "an easy way out" of difficult situations, but I think that this might be a little different. It sounds like it is not just a difficult last name for a girl if your older son doesn't seem to care for it and felt that your younger son might have benefitted from your name as well.
I do think that this is something that eventually your daughter could overcome, but if it's causing this much upset for her and your husband is ok with her changing her name, then I think that maybe you guys should talk about it as a family and decide what everyone wants to do - ie, if you change your daughter's last name, would you change your youngest's name as well so that they would have the same last name?
I can see where a daughter would want to have the same last name as her mother, too. My best friend kept her name when she got married and each of her children has her name as a middle name and her husband's name as a last name, but it has been interesting at school functions and what-not, especially with her daughter (the oldest is getting old enough now to recognize she and mom don't have the same last name, but that her friend's and their mom's do).
I think that if you decide to give your daughter some relief and change her last name that if you make a fairly big deal out of it... many family discussions, even having your daughter write out some things about it... have some kind of "process" for it so that it seems like the big change that it is, that will mitigate the potential problems with her feeling like there is an easy way out of uncomfortable things. Or from the feeling like family associations and heritage are something to be changed like shoes.
I hope that all made sense.... I'm at work, so these are just the thoughts off the top of my head.
I will say that I was moved to answer because I had an issue when I was about her age (I was 8) that my parents felt would "toughen me up" by just making me go through it, that, as an adult, I can evaluate now and while, yes, I suppose that in some ways I did get thicker skin over it, I'm not sure it was worth the cost of the wasted years and lasting bad feelings.
It's hard as a parent, though, because you don't have that crystal ball which will show you the outcome of both paths so that you pick the best one!
Best of luck with this.
P.