I’d see how most people refer to it and name it exactly that.
“The Coffee Shop on the first floor”
“The Coffee Shop by the lobby”
“The Coffee Shop at the end of the hall”
That’s life-ist.
OK, call it “Well Bean”. 
Beanie Baby
Perking Up
Because if anyone needs a pick-me-up, it’s a sleep-deprived new mom.
Will you have tables? Baked goods? Sandwiches? Not criticizing “Joe To Go” per se, but if I see that name it tells me you have a counter and you sell coffee. Here’s your latte, goodbye. If you serve food and/or have tables, it’s probably not a great name.
In a hospital environment (assuming the hospital is only leasing for one coffee shop) the audience is fairly captive, but not entirely. I’m sure there are cafes in the neighborhood. I would aim for simple and descriptive. If you want to tell us more about what your plans for the shop are, you may get some different ideas. Best of luck, in any case.
Recovery Room ------- works well for a tavern as well.
Dollar Coffee
R. Dollar, Prop.
Palpitations
Relaxation Station
Bean A Life
An Apple A Day
(I know…too esoteric. But it’s the first thing I thought of).
SVT
Code Brew
Clean Caffeine
This is the main issue.
Don’t even think about trying for a cutesy name and classy atmosphere–that’s for coffee shops with fancy garden plants and nice sidewalk seating, where people choose to spend their time.
Your coffee shop is in a place where nobody wants to spend their time.
Use a name which is simple and direct, such as the “the Coffee Corner” suggested upthread.
But don’t rely only on words–use photographs also. The sign should focus mostly on the graphic–to draw attention. Many customers will not realize you exist, and do not expect to see a coffee shop at your specific location(unless you are in the lobby near the main elevator. But if you are on the third floor between the xray lab and the nurses station, you need to draw attention to yourself.) The sign should be a photograph of the shop with 2 or 3 people sitting at a table drinking coffee and smiling at each other. The name itself is less important. “Coffee Corner” or whatever should be in a simple font, and the text should be smaller than the faces.Smiling faces make people feel good. And in a hospital, that’s critical.
Hire a professional graphic artist/photographer. You need a sign which will attract attention, and draw in customers who might otherwise ignore you.
I like single word names that evoke an idea of the product:
Brew
Up
Cup
Blend
Origin
On the other hand, it’s the hospital that I generally don’t want to spend my time, so if you can remove me from the experience by putting me back in a cafe with fancy garden plants and sidewalk seating, I’ll spend more time (and likely money) there. If I’m faced with a choice of the sterile hospital cafeteria or a cafe-like atmosphere to wile away the hours waiting, it’s probably going to be a comfortable, warm cafe.
Java Bypass
Intravenous Java.
^ Both of those are amusing.
But then all the more reason to avoid the cutesy hospital referencing names.
That said odds are a hospital located coffee venue does not have loads of seating. The bulk of the business is staff and patient families getting coffee (and maybe some other foodstuffs) to bring upstairs with them. You want to communicate quality and efficiency or maybe just appeal to a female and often pregnant or newly delivered demographic. The look can scream relax and linger but the bulk of the business is not that.
Cafe Iola. Iola is a bit of an archaic female name of Greek origin that means “violet dawn” which makes a nice subtle reference to giving birth and to hope.
I agree with the KISS (Keep It Simple, Stupid) theme…so, Coffee Spot, Coffee Corner, the Coffee Cup. Coffee Joy. Coffee Love, Coffee and Cream…
Coffee needs to be in the name, IMHO