We will soon be rolling out a person-to-person payment solution/service to our customers. How do we determine that the name we select for this solution/service is safe to use? Should we be concerned if the name is already being used for other products/services offered by other financial institutions?

Talk to a copyright attorney.

Agreed. If you use a name already taken by someone else, they will most likely send you C & D and you will have wasted all that time and money branding a service you no longer can.

Have you taken the simple inexpensive step to use lots of search engines to do plain vanilla web searches? They don't all get the same results & one might kick something out that's news.