Hey Sneakers Congrats on the new baby!
I hope everyone is happy and healthy. WOW! some spot on advice already, feel like +1 everyone who already posted. The lighting in hospitals is not usually good and flash may not be good for the babies eyes or mood. So getting a lens that will handle low light is a great idea. Also newborns are tiny so you'll need to get in close (remember to get some macro shots of the hands and feet). Please update us on the model camera you have or would like to rent. (Oops! I see you already responded T4i (crop sensor)) I love primes too!(40mm 2.8f) nice little lens, and a great zoom like the Canon EF 24-70mm f/2.8L II USM is a winner for sure even though it's not a prime.
I rent from a local camera store, so I would recommend searching online to find out if you have one around your area. It cost $35 day to rent a $2,000.00 85mm 1.2 L lens for example. Also no shipping cost if you use a local camera place.
If you have not already check out the online lensrental.com site for lenses too. The same 85mm 1.2 L cost $69 to rent for 5 days, but you should also figure in shipping and insurance as well.
If you going to buy a lens the Canon EF 85mm f/1.8 USM Medium Telephoto Lens is an afforable extraordinary lens. It maybe a little too long of a lens on a T4i (crop sensor). You can also rent a full frame body or you may want to also try using extension tube like a 12mm to enable the your lens to focus more closely than its minimal focusing distance. Kenko makes some pretty good ones and can rent them as well. It is really good on my T2i and Great on the 5D Mark II. As the baby gets older and faster this lens will be on your camera all the time. As a new born you might also look at getting a monopod or tripod to allow for slower shutter speeds.
For newborns maybe think about getting a few props as they tend to sleep a good deal and you can still get some creative shots like the one in the link below...
Here is the link to a online rental company..http://www.lensrentals.com/rent/canon/lenses
Lot of helpful videos online too.. Here is an example. She uses a 5D Mark II and a 50mm 2.5 macro lens.
Newborn Photo Techniques & Strategies, Part 1, Julie Klaasmeyer, Photovisionvideo