While juicing a lemon isn't all that difficult, below are a couple of tips you can use to make the process easier and keep all that lovely juice from going to waste:
Microwave Citrus for 10 Seconds Before Juicing This process is similar to rolling the fruit around before squeezing. Microwaves can pass all the way through the fruit, "energizing" the water molecules stored in the flesh leading to more rupture, hence more juice. It also makes the fruit softer and easier to rupture once you slice it open and start squeezing. Freeze Your Citrus First, Then Microwave It This added with the first tip will make it super-easy to squeeze when you cut it open. The idea is that once the water freezes and turns into ice, it'll expand and break those membranes once you stick it in the microwave. Your fruit won't spoil in the freezer, and you do get a lot of juice from this method. The downside is that it takes time to freeze a lemon all the way
through, and then you have to microwave it for at least 30-60 seconds until it's soft enough to cut and squeeze.
While none of these methods will magically transform your fruit into a juice sponge, hopefully it'll keep you from throwing away a half lemon that still has plenty of juice in it.