Conflicting stakeholder needs

Remember, the PO doesnt have to say yes to everything, as the PO has a vision for the product; however, the PO could devise little experiments to gain more information about them. 

What techniques have product owners found to be useful in prioritizing stakeholders' conflicting needs? 

Prioritization isn't everything. The Product Owner orders the Product Backlog to optimize value, which might be done by priority. Other things to consider include risk, relative size, dependencies (business and technical), ROI, and any work already underway.

The Scrum Guide doesn't contain "priority" or any variation of it.  It states the following

The Product Backlog is an emergent, ordered list of what is needed to improve the product. It is the single source of work undertaken by the Scrum Team.

And this

The Product Owner is accountable for maximizing the value of the product resulting from the work of the Scrum Team. How this is done may vary widely across organizations, Scrum Teams, and individuals.

As @Ian Mitchell points out there are a lot of considerations around ordering the backlog.  In the case of "conflicting needs" it seems like a Product Owner should weigh the one that provides the most value over the other.  Or which one could be done faster? Or which one will allow for the fastest feedback?  Or which one will improve the companies standing in the industry?  I can go on and on but I think you can get the picture.  There is no right or wrong answer to that question.  But the responsibility of the Product Owner is to maximize the value of the work done by the Scrum Team. In the end the organization has to respect the Product Owner's decision and the Product Owner has to be transparent in why the decision was made.

You asked for what people have found to be useful.  For me it has been the willingness to order an item that will generate goodwill over money.  It has been holding true to the Product Goal.  To quote Spock from Star Trek ... "the needs of the many outweigh the needs of the few, or the one".  (Sorry got a little nerdy there).  That quote can be altered slightly to "the needs of the many outweigh the needs of the noisy".  Also, that sometimes fixing technical debt provides more benefit than adding more functionality.