What is the best method for gathering functional requirements?

There are a number of key steps to gathering requirements.

The first is to agree on the scope of the effort. Generally, the scope can be described as:

business processes affected

Stakeholders affected

systems/applications affected

Usually a process analysis is done. This involves documenting the "current state" business processes (how things are done today), then document the "future state" of each busness process (how the business process will change). The best way to document business process is via use cases.

Requirements statements can be formed by examining by creating requirements that describe the difference between "current state" and "future state" business processes.

In some cases, where the business process is new, and there is no current state, sometimes it is effective to form requirement statements first. Then, use cases are created based on the contents of the requirement statements.

Examine your requirements statements, looking for "noun/verb pairs" and turn those noun/verb pairs into use cases. Examples of noun/verb pairs are "Quote interest rate" or "make loan" (using a banking example).