She can apply for a Family Visit visa which in general are more successful (and I think you can appeal if refuse).
But what people have had problems with is not mentioning that they are in a relationship with a British resident when applying for a Family visit visa. The problems usually occur when they apply for a fiancee/spouse visa and they state that they were in a relationship prior to their visit to the UK. Obviously she could mention her relationship with you when applying for a family visit visa to get around this problem.
If she comes here on a visit visa (Tourist/Family) she will have to leave before it expires, there is no was she can extend it beyond the 6 months. Obviously to then return to live here you will have to get married either before she come to the UK or within 6 months of her arriving here.