Have you ever wanted to listen to a TV show or a song in a loud area and you wish you could just turn up the volume so that you could hear it better? In reality, what would happen is that you'd turn up the volume and everyone would just start talking louder along with it. Not only would the volume increase of what you want to focus on, but all of the noise in the room would increase as well. You wouldn't have solved your issue at all.
That idea is sort of the way you should think about strengthening your intuition. The way to increase the power of your intuition isn't to raise the volume of your intuition but to quiet the noise so that you can hear it better. So, what is the noise I'm referring to? Well, it's the noise in your head. I'm referring to a focus on negative thoughts, stressful situations, worry, anger, obsession etc, all things that cloud your mind and increase your anxiety. Anxiety is noise; it's the loudest noise of them all.
The easiest way to quiet the noise is by meditating. First thing in the morning wake up and focus your mind on a sound, not an interesting sound, but a mundane sound like your significant other breathing or snoring. You can focus on the sound or your air conditioner humming, or cars driving down the street any boring-consistent sound. Gift yourself 10-15 minutes of complete focus on your chosen sound. Try not to use this time to speak to yourself mentally, you do that all day long. Instead, use these 10-15 minutes to rest your mind before a long day. When you first get started you'll notice how difficult clearing your mind can be. Maybe you'll go 30 seconds just focusing on your sound until your to-do list pops into your head but that's okay. That is 30 more seconds of clarity than you had before. Eventually, you will be able to work yourself up to that 10-15 minutes of a clear mind.
Pro-tip: thoughts that repeatedly disrupt your meditation over time are probably issues that you need to release.
Next, do things that help you zone out, like cleaning, reading, laundry, music, movies, journaling...anything that when you do it your mind automatically mutes everything going on in the background. Once everything is muted, then you can hear your own intuition better. Your intuition could be sending you great ideas, solutions, or warnings about situations or people. Your intuition will flow thoughts to you like it had been all along, except this time you will have turned down the noise.
Here is another pro-tip. The moment your intuition kicks in either stop and take action or take a note. I keep a journal for when I'm at home and have the time to just sit and think. Otherwise, for context, I write down what I was doing when my intuition kicked in, and the solution, idea, or warning that it offered. When I'm on the go I keep a synced reminder list on my iPhone called "inspired thoughts/actions" just in case it's an awesome idea or awesome revelation but not one I can act on in the moment.
Now go forth and use meditation as a tool to strengthen your intuition.