Okay, so maybe “weapon” is too strong. Let’s use the word “tool”.

As a project manager, leader, or in everyday life, your influence is a tool you use to get results. In many areas of your life, you don’t have authority, control, or …, and you have to nuance others to action.
An example is getting a nice table at a restaurant, or to get a table more quickly. You ask the person at the door to help you out. Of course, you’re the same as every other customer. And the person gets this request frequently.
So, you have to find some influence. You could try a compliment “Nice shoes! Where did you get them?” You can try conversation, such as the honest and obvious “You must get a lot of requests for special treatment. How do you handle them?”
This type of influence is getting them to like you, or at least interact with you from a personal perspective. There are many ways to understand influence, and to find strategies to identify, understand, and use them more effectively.
There has been an increasing amount of attention on influence. The 1998 book “Influence: The Psychology of Persuasion” by Robert B. Cialdini was (and may still be) a milestone reference in understanding influence. The core idea was the concept of the six weapons of influence. These “tools” are reciprocation, commitment/consistency, social proof, liking, authority, and scarcity. The expansion of the ideas in the book continues today.
However, the explosion of interest has come more recently due to online social websites for networking and interaction. Influence has become more quantifiable. It can be seen, measured (to some degree, in some cases), and studied. Social media sites and services like Facebook, Twitter, Google+, and LinkedIn have various ways for people to influence others. And mobile apps are just starting to enhance this trend.
How does this help you solve your project management challenges?
First, keep your focus on the positive aspects of persuasion. We know that too much persuading can be seen as annoying, pushy/aggressive, or begging. Using some of the “six tools”, having people like you is helpful, even if it is only at a level of professional respect. This only happens if they know something about you. In fact, not knowing about you makes their interaction less personal, and perhaps less of a priority.
Reciprocation is another tool that applies to management. You can divide this into these three phases of time: before, during, and after. If you do something for someone (in the past), then they will want to reciprocate. If you are doing something for them now, then it is similarly an optimal time to ask for reciprocation. The future-based strategy would be, for example, promising now to do an action, and then asking for a reciprocal favor or action. Brain and mind studies have shown that reciprocation is part of human biology, and you can do more research in how this works to understand both sides (i.e. how others may be using it).
Another tool that is not explicitly one of the six weapons listed above, is communication. People understand what is communicated to them, so this helps your influence. Additionally, people react with more interest to the ideas they have been exposed to, and the more often, the more aware they are of the concepts. For example, if you include more people in your project updates, then over time they will have seen some of this information more than other projects.
“We control fifty percent of a relationship. We influence one hundred percent of it.” – Christopher Marlowe