Make the news as big or as small a deal as they need
A lot of support during the coming-out process isn't about thrusting LGBT pamphlets on your friend, or introducing them to new lesbian friends, or telling them about the great gay bar you once went to. It's about letting them express how they're feeling and what the experience is like. Being a good friend to a person at a vulnerable time of their life, even if it's not an experience you understand, is completely possible.
Don't focus on yourself
This is not the time to tell them about your amazing uncle Ben who came out in the '80s and has had a delightful husband for twenty years. If they ask about the gay people in your life, go ahead and tell them — but until then, keep your trap shut.
Don't make stupid jokes to diffuse
This is also not the time to ream them out for not telling you beforehand or anything else insensitive to the moment.
Remember to keep your social boundaries in check. Excited or curious enquiries about their love life, sex life, how their family reacted, and other intensely private things are no-nos. You might be curious, but if it wouldn't be cool for straight friends, it's not cool now, either.
Manage your shock (or lack thereof)
If you are completely floored and flabbergasted, or if you've known for 10 years and are feeling smugly vindicated, stow it. Try to avoid gaping like you've just discovered a new species, or making 'I knew it' remarks.
You've been involved in the intimate details of this person's identity and personal life, at a massive and sometimes very scary time. That's a huge honor. Thank them for it.
Ask them about their journey
If they want to talk about how they got to this point, ask sensitive questions, like 'So how are you feeling?', or 'How long have you known?' Questions like 'When did you turn gay?' and 'Why?' are off the table, for what should be obvious reasons.
Be their straight wingman
Don't mistake inexperience with confusion
Validate their identity
Make sure you know the correct terminology. Gay, lesbian, queer, bi, asexual, nothing in particular: there are a whole heap of labels out there, and making sure you know the right one for them — as chosen by them — is a good way to be supportive.
Ask what level of openness they'd like
This is a huge part of support: keeping the information within their comfort zone. Do not tell people you haven't been authorized to tell. It's rude, and can also, unfortunately, be dangerous.
Offer resources (if they ask for some)
If they seem to be struggling, have some resources on hand — on-campus LGBT collectives, help lines, online support communities, even TV shows — that they might appreciate. Use your judgement on this one, though, and don't force that rainbow flyer down their throat.
(Oh, and one last thing — if you're feeling confused about your identity yourself, now is not the time to hit on them. Seriously.)