We arrive on a Saturday night at midnight in Quito airport (on Easter Saturday April 19th) We are being picked up in Quito city (venue can be flexible) at 6.15 am
How can we safely get to the city and where can we safely hang out until we get picked up?


Unless you want to go clubbing with your luggage you might as well just stay at the airport for a few hours and then either get the express bus to the old airport around 4.30 am and get picked up from there, or take a taxi into Mariscal or wherever in time for your pick up.
The airport itself is open 24 hours, though I'd imagine pretty much all the shops etc will be closed. But it'll be safe and you'll at least have access to toilets and (flakey) wifi.

Yes, I concur with the above answer. Stay at the airport. Going into Quito proper isn't a good idea unless you want to get a hotel room for the night. The cafe on the lower level near arrivals is open 24 hours and has wifi. From what I understand the AeroServicios bus to Quito runs every hour through the night until 6 am when it goes to every half hour during the day. I spent most of a night in this airport in July 2013 as a solo female traveler and was really tired but overall safe. You can nap if you do it by the cafe (people will be around).
I agree too, now that the airport is 24 hours it will be your safest bet if you don't want to pay for a hotel in Quito for the night.
The bus service is around $8 per person and drops you off at the old Quito airport, or you can book a cab for around US$25 (depending on where you are going to within Quito) to get directly to the destination you need to. Both services are reliable.
Where will you be travelling to the next day - depending on where you are going you may not need to go into Quito to be picked up?? Check with the people who are picking you up - tell them you will be in the airport, maybe it will be easier for them to pick you up somewhere 'out in the valley' (where the new airport is) - just a thought!