Just don’t call it “The Big Drip” if it’s next door to the Urology Department.
I like the Delivery Room.
How obvious from a distance is it that your location is a coffee shop? I think I’ve seen some examples in hospitals that really weren’t obvious unless you were right on top of them, because they tended to be tucked in a corner with windows and doors that matched the institutional nature of the rest of the building. In that case your signage might be important and I’d forgo the “cute” names for something that explicitly has “coffee” in it.
Also, being listed on the hospital directories and maps as “The Delivery Room” sounds like the setup for a terrible sitcom. 
Not as long as you have extensive midwife cross training for your baristas.
The nature of the location means there will be a wide range of clientele and a lot of first and one-time-only customers. For reasons like that, I would stay away from anything too cutesy or esoteric. Also, people will, unfortunately, be dealing with grief or loss. That’s another reason to stay away from anything too cutesy.
You need a name which is immediately understood as a coffee shop. I would suggest that the word coffee be in the name, like “The Coffee Drop” or “Coffee Cure”.
Heartbeat
Pulse
Expectations
Frank’s
Great advice!
I was coming back to add something similar to this, so “+1”. Hospitals can be highly charged emotionally even in the best of times, so neutrality is probably best. Also, some people are kind of queasy about medical stuff. Personally I’d prefer not being reminded about needles, so “drip” would be a minor irritant. A family member coming down for a break after a highly charged emotional conversation regarding to have or not have an epidural might not enjoy the perceived flippancy of having it in the name.
Honestly, it just seems like a minefield of dealing with people not at their most rational moments. Your biggest draw is that you have coffee, and maybe that you’re a small moment of escape from a stressful time. Trust me, many people in a hospital really want those things. I’d concentrate on being easy to find (“coffee” in a name that isn’t otherwise tone deaf) and providing a nice, non-hospitally place to sit for a moment. Oh, and actually making good coffee of course. 
Ulcers to Go …
Coffee Here
Coffee Rest
Coffee Cafe
The Coffee Well
You Look Like You Need Some Coffee
Coffee For The Living
Jitters
So “Coma Awakening” is right out, then?
Along with “Resurrection Coffee”
Vitamin Bean
Wake up aid station.
What’s good about this is that employees and guests will quickly start calling it “The Bean”-even easier to remember.
I’m imagining a hospital hallway with signs pointing to various departments, along with one for the coffee shop. Any name that can be confused for a medical department/facility, like “Delivery Room,” is probably not a good choice. The name should make it clear that it’s a coffee shop.
Thanks a Latte