My argument is somewhat different, BunnyKillBot, being based solely on the Christian bible (since the point is to talk to Christians about their religion).
A quick summary:
Top priority for how a Christian should view something is the teachings of Jesus, obviously. Who said nothing at all about homosexuality. So a Christian could skip the rest and just say "If it didn't bother Jesus, there's no need for it to bother me."
Next priority is the new testament. There are 3 segments in it that are currently generally interpreted as condemning homosexuality. All from Paul, whose writing is noted as being difficult to interpret. Unless you go with the idea that the Timothy segment was written a century later and wrongly attributed to Paul. Anyway...all three interpretations rely on very dodgy translations or outright fabrication.
Two of them (Corinthians and Timothy) hang on the word 'arsenokoitai' (transliterated Greek) which has no known meaning. Over the centuries, people have "translated" it into different things, essentially just whatever they didn't like and wanted their bible to condemn. ~100 years ago, for example, it was generally being "translated" as 'masturbators' because at the time there was raving hysteria against masturbating. Nobody knows what the word actually means. Having a guess based on probable roots doesn't help - you get something along the lines of "bed men", which could mean all sorts of things. Assuming that the etymology was actually Greek and the split does go where that tentative etymology puts it.
The third (Romans) is taken out of context, is unclear even in the translations designed to condemn homosexuality and has some dodgy translations. For example, the Greek words translated as "unnatural" don't really mean that. They're used elsewhere in the bible to refer to things such as long hair or Jews and Gentiles being together peacefully. "unusual" would be a better translation. Besides, the whole passage is phrased in terms of people doing something they wouldn't normally do, apparently in connection with something ritualistic for another religion. So it simply doesn't apply to homosexuality in general.
Next priority is the old testament. That's a more complicated argument over translation (e.g. "abomination" is not a good translation of "to'ebah'") due to the greater differences between ancient Hebrew and modern English, but it can all be ignored unless you happen to find a Christian who can give an internally consistent explanation of which OT rules Christians should obey, which they shouldn't and why. Even if the OT does condemn homosexuality (which isn't clear), it equally condemns numerous other things that Christians are almost always unbothered by, such as mixing meat and dairy (e.g. cheeseburgers, lasagna, etc), wearing clothes made of different fabrics, planting two or more types of plant in the same field, etc, etc.