Chickens, it was good that you started this thread. I think within our culture, there is a tendency to disbelieve or at least insist on a much higher level of "proof" in cases of gender-based violence or abuse. Several others have set out the context of institutionalised sexism that contributes to this.
Because of this, people will always ask, "but what about the false claims of rape," and it's important not to say anything that would collude with the myth that this happens alot. However, if we can provide an explanation, and the context for those rare "false claims," then people are more likely to engage with us, and let go of their disbelief than if we just try and brush them off with, "it's so rare, it's not worth talking about." It just needs a lot of care.
It's important to be clear that even of a person isn't convicted of rape, it doesn't mean someone lied (any more than it means a person lies about any crime where someone's not convicted, like mugging, arson or fraud.) Point out that "popular papers" don't print stories about rape that reflect the actual incidence in terms of "false claims," and often include cases where there was no conviction as "fake."
I think it's important though to be careful about how we seek to "profile" women who "do" make "false" claims of rape. As others have said, there are situations where they've been raped, but accused the wrong person due to mistaken identity. There may be those who say they've been raped because they believe this would be more acceptable, say to their family, than to say they'd had consensual sex. That would suggest they have lied to avoid some kind of "punishment," which in itself is not right.
As a few others have said, many of those likely to make false accusations are also likely to make false accusations about other things, perhaps because they don't have the capacity to get what they want or need in life through communication and negotiation. Sometimes, they will have been "taught" to get what they want in need this way from parents, peers or abusers.
Be very careful though of saying that those who make false claims are likely to be mentally ill. It is true that some people who are mentally ill will have delusions or make false claims about all sorts of things as a part of their illness. However, let's not forget that there are many, many people who experience mental illness, and ARE raped. In fact, abusers may "home in on them," because they are aware of their vulnerability AND the likelihood that they won't be believed (so they'll get away with it.) It is important not to inadvertently suggest that people with mental ill health are more "prone" to false claims.
There was another point upthread that bothered me alot when I read it - about the 13 year old girl who said she was raped but was actually "hanging out with drug dealers," or similar. If she had sex, even with "consent," it's still child abuse, still rape. She may have been told by the dealers to say she was raped to keep them out of trouble, or more likely, threatened her if she did anything to implicate her. Chances are she'll have exchanged sex for food, drink, drugs, phone credit, etc., and that chain starts with child abuse, usually sexual abuse. Too often, we DON'T see situations like that as child abuse, but we should.