Is there a way to check whether an iphone application name is available to use? Would a search on iTunes and checking to see if there is an existing application already out there be indicative that an iphone application name is available to use or not?
I would recommend searching the app store for the name you wish to use. Also, you might want to check the United States Patent and Trademark Office www.uspto.gov/ to ensure there isn't a trade mark that exists on the name you choose.
I'd say if the name you want to use doesn't turn up in a search on iTunes, you should be in the clear to grab that name. I assume all app names are subject to approval by Apple, also, so make sure you pick something that's not called "iBabyShake."
As pointed out here, even if you search the traditional channels, there may still be someone who has claimed that same name but not submitted an application yet. If you wish to truly test this out, it sounds like you can start the process of submitting an application on iTunes Connect, fill in the name of your new application, but don't complete the submission. You should receive an indication as to whether someone else has this same store name. If not, it appears you now have a claim to that name.
In any case, because you can have a different name in the store than is displayed on the iPhone, you might be able to use a slight permutation on the name for your store submission, and your desired name within the iPhone application itself.
However, I would second sgmeyer's suggestion that you do a USPTO trademark search first before using a name. Trademark infringement can get you in trouble later on.