So there's one particular girl at my gym. Def catch her staring at me every now and then quite a bit, but she keeps to herself. If we walk by each other, she has tunnel vision and doesn't make eye contact.

As a big workout guy myself, I don't really approach girls at the gym. I personally hate to be bothered at the gym, so I afford girls their space their too. Ironically, a decent amount of girls have approached me at the gym, but none which I'm interested. Point being, I rarely initiate convos unless it was obvious and easy (ex: I see a girl studying for same tests I studied for on elliptical, its raining and we're both walking through the door same time, bs like that).

So... how to approach this girl at the gym? Any openers? Common ground ideas? Literally stumped!

Yeah...

"Hi, I'm MN87" -- and smile.

Then say you've seen her around here a few times and wanted to say hello.

Seriously. You don't need a line. The above is the best there is.

Ditto, you're making this way too complicated. Next time you get any eye contact or next time she isn't on a piece of equipment, go up to her. Tell her you're busy working out but you've seen her around and wanted to say hello. Introduce yourself. If she smiles and is friendly tell her that you don't want to ruin ya'lls workout and that she should give you her number so you can get in touch later.

If she has any interest she'll say yes. If she's very eager then make plans right there.

"I know we're both busy working out and I don't want to take up too much of your time, but I think we should go grab a cup of coffee this Saturday, give me your number and ill text you."

This is probably exactly the time you need to say something. Even just Hi or something like CharlesFinley suggests.

Think of it this way, she's probably shy and doesn't want to be too obvious when walking past you.
Also, with other girls approaching you... which ways did you prefer to be approached? You may not have been interested in those specific girls, but you can certainly mimic their technique if you thought it was good.

I dunno about that tbh... she's actually a professional dancer and has was a professional cheerleader for some sports teams.
Really interesting point. I feel like it's slightly different with girls though. They don't need to worry about the approach, because most guys won't reject a girl I feel.

There's a difference between being in front of a million people and opening yourself up to someone you like...
Not really the point... if YOU liked what they did, it's probably a good way to go. That said, women rarely if ever approach and society sometimes tells them they shouldn't... so even if we logically think they "don't need to worry" it's likely that they are anxious about it. It's not like they've gone through the newbie mission you know...

I can relate with you MN87. Cold approach in a gym is tough. I also enjoy weight training, and I find that when I'm "in the zone", and I'm referring to the weightlifting zone, not pick-up zone, I really don't want to be spoken to. But you know what that zone is? It's my comfort zone. I'm comfortable being in my head, pushing the boundaries physically. Cold approaching girl is out of my comfort zone. This means I have to get out of my little training bubble, and do it anyway.

The girl is probably there to make herself more attractive to the opposite sex, and would most probably wants to be approached by you, hence the eye contact. Also, the eye contact makes the approach a little warmer than a solid cold approach.

"Hey, I don't usually do this, but I wanted to meet you."

You can then shake hands, and move forward with the interaction.

This works a charm. You have validated the girl as one of the few girls you actually approach...And think of it like this, you've not only trained your body this workout, you've also trained your mind.

It doesn't need to be difficult at all. Just make an offhand comment to her about something going on and vibe on that for a bit. I.e. she gets done with a particular workout. "I haven't tried that machine what's it like." She just wants to feel your energy. Give her some of it.