Vegetarian - Westside
Maybe go for brunch? She could eat french toast, eggs, or pancakes. Any Indian place will have lots of veggies choices, and any italian restaurant will probably have several pasta dishes. If you go for brunch, Hal's in Venice has lots of good vegetarian choices: http://www.halsbarandgrill.com/pages/food__spirit/brunch.php
If you go for Italian, Fritto Misto in santa monica is always a good bet for casual pasta:
And there was just an extensive thread on Indain restaurants if you choose to go that route.
in westwood, for dinner, i'd go to the persian restaurant, shamshiri on westwood blvd. they have a large portion of their menu dedicated to a variety of vegetarian dishes. also, the meat-eaters will find plenty to eat there too.
for thai food, i'd go to ayara thai cuisine, which is located near LAX. the whole portion of their menu entitled, "from the wok" can be ordered vegetarian, made with tofu.
if she's open to ethiopian, there are a couple of excellent choices on fairfax blvd, my favorite is rahel which is ethiopian vegan; they have the very best injera. i also like the somewhat more upscale and definitely more spicey meals by genet, which has a lovely vegetarian plate.
OP asked for lunch...unfortunately, Meals by Genet's Website (http://www.mealsbygenet.com/) lists their hours as 5:30-10pm Wed-Sun. So, no lunch there - as I discovered to my chagrin this past Saturday. Ended up at Rahel and while I appreciate the vegan-ness, it was a disappointing meal, especially the injera. Not at all fermenty tasting like the best injera. I had been to Rahel before and I remembered the quality being higher overall, so maybe it was just an off day???
I'd probably recommend either:
-Annapurna for Southern Indian. They have a weekend buffet, but they also have an express menu that includes dosas and that makes a nice brunch/lunch.
-Tender Greens, for salads or grilled veggie-based sandwich or platter. They open at 11:30, I believe.
Both in Culver City.
I have been afraid of Rahel....it's like...I always say I am only as vegan as the last time i had ethi food...because i cannot IMAGINE my food made without niter kibe (spiced butter) i mean im sorry it runs in my veins! I NEED IT hehe...but, i am always open to seeing how the cook has managed to make a good misir wot without the real deal good stuff!
You shouldn't worry so much. Vegan, maybe a little difficult. But almost any restaurant you can think of (in LA, anyway), is going to have some sort of vegetarian option, and most high end places will happily do a vegetarian plate even if there's nothing veg on the menu. If you haven't already, it might be a good idea to try and find out what type of food she likes (or doesn't like)
And this is probably common sense, but try not to make a big deal about it to your SO's daughter - most veg*ns don't like to feel as if they're putting someone out or causing undue stress on someone else, and we're also mostly used to dealing with less than ideal situations when eating out. If she ever comes over for dinner, this is extra true - even if you're not used to cooking vegetarian dishes, I'll bet there are a ton of things you make right now that are vegetarian / vegan, or could be made that way with little effort -- so don't make a big deal about how hard it is to accomodate (my SO's parents, who are Chinese, make vegan food all the time when I'm not around, yet somehow they always get stressed out when I come over for dinner).
Anyway, back to your question....
Italian's almost always a safe bet; you'll hear a lot of good stuff about Angelini Osteria on this board. There are a couple Ok (not amazing) Italian places on Westwood Bvld. south of Wilshire.
With Thai (mentioned above), be very careful to ask for no fish sauce, shrimp paste, or oyster sauce. I personally think that if you're vegetarian and avoid fish, it's better to only eat at vegetarian Thai places or fairly vegetarian aware places like Toi. A *lot* of Thai places have "vegetarian" sections or claim to specialize in "vegetarian" food, yet put fish sauce in everything.
Le Pain Quotidien (in Westwood Village) would be a good spot for a light lunch (soup, salads, sandwiches, etc.), and they have a lot of vegetarian and vegan options. Ethiopian (also suggested above) or Indian are also good options, at least if everyone likes that kind of food.
Native Foods (also in Westwood Village) is a vegan place that's great, if you don't mind going to an all vegetarian place.
gr/eats (Giant Robot's restaurant down on Sawtelle) is decent if not "gr/eat", and has some good vegetarian options.
I'd suggest La Cachette, but they're not open for lunch on weekends (and I wasn't blown away when I went there anyway).
We (two of us) had a real good vegetarian lunch at Border Grill (4th St. in Santa Monica) yesterday. It was my first time going. The best thing on the table was the "Border Vegetables" entree, which had about nine different hot and cold vegetable preparations (mushrooms, yams, corn pudding, quinoa salad, grilled cauliflower, watercress salad) arranged on a big platter. We split that and the grilled mushrooms stacked with peppers and greens atop a sort of chili-tomato sauce, which was also good. And the service was very attentive.
There were other vegetarian entrees to choose from, and I'll definitely go back. We left quite full.