My response: I think the best way you can help this young lady to accept you as a friend (and not as a replacement for her dad) is simply to be a friend to her. In other words, it's all in how you behave toward her, regardless of how she feels or behaves in response to you. That is a tall order, I know, and if you find over time that this is beyond your ability to accomplish by yourself, you might consider seeking professional help at some point. In the meantime, however, all I can suggest is a lot of patience and understanding on your part, since you are the adult in this situation, and you are dealing with an adolescent who (if she is like most young poeple her age) is struggling with her own identity and the (normal) developmental tasks of discovering herself, turning to her peers for understanding and support, and separating from her parent(s). If that were not enough, she's also dealing with the grief she feels at the loss and absence of her father ~ at a particularly vulnerable time in her life.
It may be helpful for you to keep in mind that the people in this family each are grieving very different losses, and the relationships you all had with the person who died are very different too. You lost a dear friend, your lady lost her spouse, and this young girl has lost her father. However she reacts to her father’s death depends on many, many different factors, and her reactions will change over the years, just as she will change and develop into an adult woman.
In her insightful book Fatherless Women: How We Change After We Lose Our Dads, author Clea Simon observes that even mature, adult daughters of the newly widowed sometimes have trouble “balancing the real vulnerability of our newly single mothers with respect for them as adults.” She goes on to say that
Accepting and encouraging our mothers’ independence can be awkward for us . . . Particularly in the social arena, we are not usually accustomed to seeing our mothers as women. We knew them as our mothers, not as fellow adults who raised us, who worked in the house or out to keep a family together. We do not usually picture them as women like ourselves, as partners enjoying or leaving relationships, as people like us who have lived with the mixed consequences of their actions. Unless our mothers had been alone for a long time before the death of our fathers, we tended to see them as part of a unit, as teamed with our fathers (or stepfathers or partners) in their roles as our mothers, not as women. Now fate conspires to show us the other faces of our mothers, and makes this time full of discovery for us both. For many of us, this can be an uncomfortable transition. If our mothers start dating, for example, we have to accept them as sexual beings. If we have not faced it before, we are now confronted with the reality that the tight parental unit – the monolith of parental support, discipline, and security that protected our childhood – was comprised of two humans, one of whom is now single and lonely as we have ever been. Some of us may experience this discovery as a betrayal . . . After the death of a parent, particularly a father, this . . . may become most pronounced when a widowed mother becomes sexually active again . . . (pp. 140-142).
I think in situations like this it is always good to read and to learn as much as you can about the normal grief process in general, and grief in adolescents in particular, so you'll be in a better position to understand and deal with whatever reactions may come up for this girl. See, for example, some of the articles and resources listed on the Child, Adolescent Grief page of my Grief Healing website. See also the articles listed here: Children, Teens and Grief.
You might also find this article helpful: How Long Is Long Enough? by Julie Donner Anderson.
I'd also encourage you (and your widowed friend, if she is willing) to consider the various suggestions offered in articles such as these:
Four Things a Widowed Parent Should Know About Dating Again by Kimberly Ramsahoye
Dating when you have teenagers via Widowed Village
Although dating after a death is certainly different from dating after divorce, these articles offer additional insights that may be relevant in a situation like yours:
Children and Divorce - American Academy of Child andAdolescent Psychiatry
Dating After Divorce: What it Means for Kids by Katy Abel
Dating After Divorce: How soon is too soon for the children? by Lisa Zeiderman