It can be more than an extra paycheck.
Much more in some cases

A part-time job can be a great idea once you have all the other aspects of your financial situation organized.

Before you consider a part-time job, there are a few things to consider first.

* As mentioned, you really need to know exactly what your financial picture already is.
It makes no sense working extra hours away from your normal job if you have no idea where your money is going in the first place.
In most cases no part-time job is going to pay you anywhere near the amount YOU WILL SAVE patching the holes in your bucket that is your current financial life.

* Don't neglect your day job. Your real job comes first.

* Speaking of day job, be sure your part-time job doesn't pose any conflict of interest in any way.

* The money will rarely be as good as full-time work.

* There's nothing easy about an extra 15–20 hours a week on top of a regular full-time job.

* Are you absolutely sure you can work that many hours and still not negatively affect your day job?

* Don't forget the tax implications!

* Despite the negatives we've listed here, There are far more positives in the majority of cases.
Including a few you haven't thought of.

* More money! It's the whole point.

* Get paid to learn new skills.

* There's few things more important than additional training throughout the course of your life.

* Additional training in a variety of fields can only help you with your primary job or career.

* It shows ambition. Every employer likes to see ambition in their workers.

* The right part-time job can help you in transitioning into a new career (see "changing careers").

* Working with a larger variety of people helps you with your networking skills.

* A part-time job can pay off debts or fund your investing goals  SO MUCH FASTER THAN IMAGINABLE.

Time for an example:
Your regular job leaves you with $100 per month extra for debts or investing.
By any measure that's pretty tight.
Add an extra $400 a part-time job brings and suddenly those debts are going down by $500 a month!
Better still once those debts are paid off $500 per month into an index fund will supercharge your investments.

It should be obvious by now a part-time job offers you more than extra money.
Particularly if you can merge it with the skills or some specific knowledge you want to acquire.

Before you do get a part-time job make sure you cover these points.

Give yourself a raise First!
More on that under the title "give yourself a raise". 
Do and redo your budget and make sure to remove any unnecessary expenses first.
Also, make sure you're not overpaying the withholding tax on your day job.
Working part-time only to cover the $300 you didn't even realize you were wasting is a sure way to never financially progress.

Designate where that (part-time) money goes BEFORE you get it.
Your probable working part-time in the first place because you need the money.
Every dollar goes where you tell it to.
If you want to use some of that extra money for something that's special to you that's fine. 
But remember why your working all those extra hours in the first place.

If possible, look for a part-time job that you will find fun.
Whatever fun looks like to you.
Maybe a job outdoors. Perhaps something working with kids.
If you can make extra money doing something you already like doing it's win-win.

One final thought.
You'll most likely find seasonal work is higher paying than a typical part-time job.
Generally seasonal work is more demanding but higher paying than typical part-time jobs.
We mention this because some people prefer an occasional better paying seasonal job over steady but lower-paying part-time work.

Above all else, be sure whatever path you take makes sense for you.
You're the one working all those extra hours.