How to ask for this letter of recommendation?

Just got with the first one. Many professors are glad to write letters of recommendations, especially if they like you. Remember, it's part of their jobs so it's not too crazy a request.

Pineapplicious nailed it right on the head.

Pineapplicious is right about recommendations being part of the job and professors expecting such requests. It's also good advice to ask whether the professor could write a positive recommendation. Since the professor is on research leave, though, and out of the country, she probably does not have the secretarial support that she would normally have at home. Hence she might not have the time or inclination to send out several letters. If your friend's college has a career center with a credential service, your friend should find out whether the service handles grad school recommendations and whether the professor could write one recommendation that the credential service could send out to every school on your friend's application list. That could save the professor a lot of hassle and your friend a lot of worry. One online service for such credential files is Interfolio.com, but there may be others, and again, it's worth checking with the college career office or center.

Another voice chiming in to say that you likely don't need to worry overmuch about being rude; professors get asked for recommendation letters all the time - heck, I'm only a TA and even I get the occasional request. Personally, I'm kind of flattered when a former student asks me for a letter; maybe your professor will feel the same way?

Yeah, don't make up BS questions. "Can I stop by to ask you a few questions" is a totally legit way to preface a request for a letter. You could ask her for advice in your application process, and that doesn't seem fabricated to me in the slightest. When you do ask for the letter though, make sure to ask if she would feel comfortable writing you a strong recommendation. I know it sounds silly, but this is almost like a codeword of some sort. Most any prof will dash off a letter if you ask them to, but a strong letter is something that they are less likely to agree to if they don't really mean it.

Professor here... if I was on the other end of this exchange, I would definitely be absolutely fine with a reply that said something like the following: "Thanks so much for your quick reply! I'm considering going to grad school in [field X], as I find that it is [insert reason for wanting to go into that field]. I had emails because I was wondering if you would mind writing a letter of recommendation for me, and was also just interested in your insights about grad school in general, since I found our discussions in office hours so useful when I took your classes. Since you're overseas and it's harder to maintain such discussions over email, I don't want to impose on that count -- but do hope that you'll be able to write a letter. [Insert logistic details of where the letters need to be sent, when they are due, how to mail the forms and SASEs, etc]." This isn't too different from Pineapplicious' suggestion, but it does provide a way to get around the issue of what "other questions" you had. I'm sure her insights about grad school would be useful, even if it's a different field -- there are vast similarities from field to field -- but writing it in this way also gives her an easy out if she doesn't have time to have a discussion about that over email. (The reason I suggest explaining why you're interested in field X is that it will help her write a good letter, since she'll be able to relate things you did to that, if there is any relation to be found). I second everybody in saying that she should be absolutely fine with writing such a letter. I generally enjoy writing such letters, especially if they are for students I knew well; and in any case it is an expected part of the job.

Many grad schools either allow or permit letters of recommendation to be posted through a secure web site, so if she can use that method it shouldn't be too time consuming even from overseas. You could find out at the websites of your target schools or departments whether that will be possible and pass that information along with your request

Whenever students who aren't currently enrolled in my course ask to meet me "to ask me some questions" I always assume that it's about reference letters. I'd be shocked, actually, if it weren't about reference letters- what else would a former student have to ask me? Just ask.

Also, tell her the deadline for the entire application. But also ask if she would prefer to see your application ahead of time, and if so, when she'd like to have it. Some professors want to have as much information as possible in front of them to write a strong, targeted letter. Others will work mostly from memory, and others, somewhere in between. By asking her in advance about her preferences, you'll make sure to avoid last-minute surprises or frustrating an otherwise valuable letter-writer. And she definitely knows how to interpret "some questions about grad school." Don't discount the idea that she might still have some words of advice about particular schools, even if she didn't go to grad school in your discipline, or teaches in some other subfield.