Great question. I think this is personal to each of us, but I can only tell you what my feelings are, and what I did.
I think it's different when there is infidelity involved, for starters. My wife (who was having an affair during the Summer of 2007) was under the illusion that somehow we were going to stay best friends even if we divorced over it. So one day -- about a month into my knowledge of her affair, and her refusal to end it -- I told her that:
"I should be clear with you about something. I have absolutely no intention of remaining 'best friends' with you if you choose to end our marriage this way -- by having an affair, running away, and lying to your parents and our children about it. We'll be civil, and we'll co-parent effectively, I'm sure, but we won't be friends. If you decide to end your affair now, however, and come back and work on this with me, going to marriage counseling, each of us addressing our issues, and it doesn't work out -- say after a year -- and we choose to divorce, then yes, I could see a time where eventually we could become good friends again, even though it won't be the same. But not what you're doing now, I'm sorry. This is NOT how friends treat each other, and I respect myself too much to put up with a so-called 'friend' who would do that to me."
She told me the day after she ended her affair, and asked back into our marriage, that this was the NUMBER ONE REASON why she decided to end it. "I missed our friendship," she told me, tears streaming down her face.
As someone stated above, however, this has to be how you TRULY FEEL. You can't do this as some "technique," to win them back, as that will smack of pursuing (and you shouldn't bluff with something as intimate as a "best friendship" anyway). It has to be authentic.
"What is best for my kids is best for me"
Persevere = happily being patient over a long period of time