I have a lot to say about this, as I am a phone volunteer at a crisis center where I routinely have to address all manner of issues, suicide being one of the most serious of them.
I'm limited by time right now, so what I will say is that, based on what I've seen fielding calls, suicidal people are largely suicidal because they are seeking relief from a feeling (or lack thereof). In large part, having someone to whom they can talk and trust is paramount. Just listening to someone actively (paraphrasing their feelings) is a form of relief in and of itself.
When I'm on the phones, I don't try to take away the option for suicide - they are feeling suicidal after all, I am not going to deny what it is they are feeling (in fact, I do the opposite and will openly talk about that suicidal feeling) - but I also sit there and just exist for them. It doesn't always help, but by and large I've found that just being there can provide immense relief. (I should add that I do more than just listen, but active listening is the number one component to what I do.)
I think that's a far-cry better than permanently removing that pain. Suicide is an option, but it isn't the only option.