First of all, thank you for giving men in the gym a good name. I've seen the husbands and boyfriends who push their SO's around, yelling and lecturing them, and then get angry when the woman doesn't want to continue with a weight training plan. Gee, I wonder why?
Secondly, good on you for being aware of what your finance will and won't do. You've just made the process loads easier on both of you. The dumbest thing ever is to tell a person what they should want to do, especially when it requires a good chunk of effort for them to do it. There are two ways to get anyone to do something and only one of them works in the long term: you can motivate them or you can inspire them.
You already know about motivation. It's when you do something because you want to avoid an unpleasant outcome. But inspiration is like lighting a fire inside someone that they keep going all on their own. It's a thing of beauty, it's a thousand times more powerful, and it's more satisfying all around.
Here's how I talk to women in the weight room who ask me questions about moves. First, I tell them how smart they are for wanting to strengthen that muscle. I work into the conversation a few ways working that muscle will really pay off for them. Like, in your girlfriend's case, I'd tell her she's a sharp lady for wanting to work arms. Arms are overlooked by most women because they don't understand how important they are or they think their arms will become huge and they'll look like a female Arnold Schwarzenegger. But smooth, well-shaped arms look fantastic in halter tops and spaghetti strap T's, not to mention those swimming suit days. Having strong arms will mean she won't be one of those helpless women who need a guy to get every box down for them or open every jar, either. And if she is inclined towards children at all, holding a baby, a toddler, and then a small child in your arms is torture for women with no muscle tone. Ladies who know their way around a gym can play WITH their kids instead of just sitting on a bench in the park and WATCHING them play. If she's a scientific mind, you can throw out a couple health benefits besides, you know them all by now. My favorite is how weight training actually increases brain function. Kind of blows the "dumb jock" theories out of the water, hunh?
OK, so, now that she's stoked, feels good about her decision, and is looking forward to the future--she's inspired--casually drop how easy this is and how the more you do it the easier it gets. Tell her weight lifting is a self-directed sport, you can start wherever you are comfortable and there's no scorecard for everybody to see. It's all about how you feel, listening to your body, and gradually teaching it to get stronger. This a sport between you and you. She is in charge. This is a liberating idea for a lot of women. Plus, it will take the pressure off you. With her in the driver's seat, it will be more like you are a friendly, trustworthy resource for her to tap rather than a coach she has to be afraid of disappointing.
Other than the usual safety measures (lift slowly and controlled, keep excellent form, when to breathe, etc.) a key point is to make it understood how muscle groups need to be worked as a team. If you aren't clear on the science, yourself, have some fun together looking through books at the library or the bookstore that show really good pictures of all the muscles that assist delt lifts, rows, wrist curls, etc. The machines in the gym have little pictures printed on the side to show which muscles are being used but that doesn't tell the whole story. The really good books will give more detail and in simple English. When it clicks that each muscle is assisted by another, she will accept the necessity of more than one or two exercises for each body part. Her arms are complex machines. Just doing curls and bench press and calling that an arm workout is like only doing gas fill ups and oil changes on your vehicle and calling it "maintained." There's a whole lot more going on under the hood.
So, for her arms to be truly strong and balanced, she needs to work biceps and triceps, forearms front and back, all heads of her delts at the shoulder, all heads of pecs, upper and lower back and her core muscles for stability while performing all these exercises, and some traps and neck work. This sounds like a lot and it is but you don't have to do it all on a Tuesday. There are combination exercises that will cover several of these in one fell swoop. Start with the obvious--biceps, triceps, delts, and back--and when she feels comfortable and confident with those moves, show her one new move every couple weeks. Make it fun and she will look forward to the weight room as much as she does the stationary cycle. Drop a couple compliments on how sexy her arms are looking and she will wrap them around you a lot more often. Woo-HOO, Eric!