Is there any way to check the availability of iOS app names?
NEW INFORMATION (2018)

Please see this question on the relevant site, Stack Overflow: 
https://stackoverflow.com/a/17748195/2684

tl;dr: there is no time restriction on how long you can reserve, but it’s not guaranteed. 

OUTDATED ANSWER

I haven’t seen that option in the Apple Developer Program, but you can reserve a name. 

After you reserve it, you have 120 days to upload a binary (You have to create an app in the iTunes Connect system). After 90 days, you receive a notification from Apple, saying that you  only have 30 days left to upload the app. Once the 120 days are past, "the app name will then be available for another developer to use.”
I also believe that if you do not upload anything, bad things happen to your account, but I can’t verify this.

Other than that, I haven’t seen a mechanism like “google” for name searching. (Apple might as well do that at some point!).