How to choose the right stage name
I believe you should treat choosing your stage name with the same consideration you would choosing your children's names! You are after all, choosing the name that will stay with you throughout your career and become an embodiment of what you represent.
Firstly I wouldn't ever advise changing your first name unless it's to something you are used to answering to because when you are on-set or in a theatre dressing room and over the relay comes the booming call "James Gough to stage please, James Gough to stage" and you're sitting there wondering what that wally is playing at, when it hits you that they are calling you and it hasn't clicked it's not a good start. So I'd say stick with your first name, or if for example your friends call you by your middle name then that is ok too.
When choosing your surname, firstly you will have to give 3 options. If a name is taken on Equity and Spotlight already then you can not duplicate it. It is for this reason that you are asked to give 3 options. Many people give their own name as a first option but if your real name is as common as mine there is a good chance that the person dealing with your application is also called that. Don't panic if you can't use your name, this is a chance for you to create you (the actor) as a separate entity to you (the onesie-wearing weekend warrior). I fully embraced this change and now I wouldn't go back.
Pick a name that means something to you. That would be my biggest piece of advice. Sure you might have seen a cool name on a beer mat but 20 years down the line that might not stop you hating it. If you pick a name that has personal significance to you, just like with a tattoo, it will always carry a little more charm. Be creative, for example mine came from the hospital I was born in but fits my casting and means something to me and my family when we see it in lights. It feels like you, not someone else up on that credit roll.
The final point would be to think about your casting. If you are destined to play the nerd don't call yourself Hercules. Although funny in an ironic way that would tire. Try and pick a name that suits where your look fits into the industry. If you've ever had someone say to you, "yeah, you look like a Laura!" this is what we are aiming for! Pick 3 but take your time. This is part of your marketing, like choosing a company logo. Give it the respect it deserves and then when you walk into an audition and someone says "yeah you look like a Michael Chase" you can smile a little sly smile to yourself and know that you've nailed it.