1. Read a new translation.
Pick a translation that you haven't read through yet. Even small differences will make the passage pop out to you in new ways. Sometimes different words help us to realize that we had assumed the passage meant something it didn't necessarily mean. My suggestion is to pick a version that is slightly more literal than what you've been used to. For example, if you are used to the NIV try reading through the ESV instead. (NLT to NIV to ESV to NASB) I'm not saying that loose translations are always bad, but the further a translation strays from the original the less helpful it is.
2. Read out loud.
Reading out loud forces you to pay attention to each word. It also causes your brain to process the information differently than if you were just reading it silently. Reading out loud also has the benefit of keeping your mind from wandering away. This can be especially helpful if you are tired.
3. Read it to a kid.
Reading the Bible to a child has many advantages! If you are a parent, this is an important way to fulfill your duty to train your son or daughter in God's truth. I read to my oldest son every night before bedtime. It is great quality time together and it helps him see how much I care about him and God's Word. Reading to him also helps me read the Bible in a fresh way because I have to explain words and think about the questions he will have.
When my first son was very young we read through The Jesus Storybook Bible several times. I recommend it. When he was in second grade, we started to read through the Gospel of Mark one "episode" at a time. This worked well because they were short, focused stories. Also, Mark was good for that age because it is action-packed and focused on Jesus. After Mark, we started reading the Old Testament one chapter a night. We started with Genesis and just kept going. We talk about applications that stand out to us but I don't try to "teach a lesson" each night.
4. Write out the passage.
This may seem tedious and pointless but it is not! Writing out the passage forces you to focus on individual words and phrases. You will be shocked by how many details you would have otherwise skimmed over. Now, you probably won't want to do this for long narratives or genealogies but for other passages it works well. It is ideal for New Testament letters! Try writing out the entire book of Ephesians longhand. I guarantee you will not think it was wasted time.
5. Write Bible study notes
This technique is not as simple as the others but you might want to give it a try. Try making a running commentary for your passage. Imagine that you were writing notes to give to a younger Christian who is preparing to lead a group Bible Study on this passage. For each verse, explain anything that you think needs to be explained. What is the main point that the author was trying to communicate for each verse or paragraph? What are applications that we can draw from this? I did this for fifteen years for small group leaders. Even when I had leaders that didn't "need" the notes, I still did this because of how much I got out of it. I'm sure it would be a benefit to you as well. And hang on to those notes; they may come in handy one day!