What trade-offs does an airline make when selecting an engine type for their aircraft?
Power

Some models are different in power, a consideration if you intend to operate hot and/or high, or heavy. Otherwise that extra power is mostly fuel-inefficient.

Competition

In situations where two or more alternatives are available, there's healthy competition. Engines are around 25% of the cost of a new aircraft, so there would be a big money to gain when airlines bulk order from a certain company.

Boeing took this a step further on the B787 where it's easier than ever to switch the type of engine to prevent this type of 'vendor lock-in'. Historically,  it was very difficult to switch due to the technical complexity in these systems.

Customer Relationship

If an airline has good contacts with a manufacturer and know how they work, they may prefer to opt for that comfort when considerable money is on the line.

Part Commonality

The more type of the same engine you operate, the better economies of scale you achieve. This is especially important as substantial maintenance work is concentrated around the engines. 

This made for instance American Airlines store Boeing 757 aircraft that it got through a company acquisition that did not have the same engine type as the remaining fleet.


This thinking is not limited to your company, but for instance within a group of companies, as heavy maintenance is often centralized.
It can even apply across types. For instance, Korean Air ordered 787s and 747-8 both with GEnx engines. To quote GE Aviation on the two slightly different engine types: 


  The -2B contains an identical core and similar materials as the -1B, delivering synergies for mixed GEnx fleets