Thanks for a good Question.
I would suggest a few responses.
Love them. What does a Christian have in common with someone living in a same sex relationship? Generally we like to love and be loved. So first thing is to love them and work to love them, even if that is not returned. Being a friend who loves them for being a person before being someone who is ‘gay’ is a good step.
Pray for them. Christians can pray and know that God is listening. Our unbelieving world don’t have the same privilege nor confidence. So a loving thing to do is to pray for them and put them before God.
Worry about the Gospel. Be most concerned with the most concerning. A good question to ask is ‘What is the most important matter here?’. It is not that they are same-sex attracted. It is not that they are in a same sex relationship. It is not that they see no issue with their lifestyle choice. The most pressing issue is that they have no gospel hope. Christians should be more concerned with talking about the good things offered in the gospel and less about the bad things that sex-sex attraction brings.
Describe a Gospel alternative. Often Christians can fall into the trap of making this matter about sin and trying to call the person to account. The Bible clearly calls homosexuality a sin (along with many others things that we know all too well), but as you have pointed out, this person is not that concerned about what the Bible says so why would they be bound to it
Those who hold or promote a Same-Sex lifestyle will often sell themselves short. They take desire, behaviour and identity and make them all the same thing. That is a shame, because there is so much more to a person that their sexual orientation and we should work to treat all people (gay or straight) as far more valuable than that determined by their sexual preferences.
Jesus loved the person – and died for them. He loved in spite of desire. He loved over and above behaviour. And he loved knowing the true identity of a person (someone made in the image of God). Jesus loved by speaking truthfully yet lovingly and in doing so holding up the persons great value, yet dying in their place. So in the gospel, there is a hope which is deeper and more fulfilling than desire, attraction, sin, or identity.
So talk about the character, words and work of Jesus and keep the topic on values that strike a cord with those who want to be loved, fulfilled and understood – because that is what Jesus did.
Raise questions rather than give answers. In my experience, those who hold to a same-sex attraction position generally enter a conversation with a Christian making strong assumptions about what a Christian thinks (i.e. not in favour, will reject, closed minded etc.) and with good reason. Christians who hold to the Bible, will disagree with the same-sex position. Therefore, you don’t need to defend your position (even if they have faulty assumptions).
But interestingly, those who hold to a same-sex attraction position generally have a myriad of competing reasons why they hold to the position. Often they are based on experiences or reasons that have unfulfilled origins. It can be helpful to ask lots of questions in an effort to understand the person and why they identify as same-sex orientated. Ask, not to change them, or convert them, but rather to love them, understand them, care for them, address past hurts, and be a friend to them. And in the process, and over time, you will probably be able to point out the conflicting tensions that are a result of their same sex position (e.g. the self-determining nature, biological childlessness, no eternal hope, value as a person being limited to sexual orientation, difficulty in making friends outside of the LGBT community etc).