<<As a goal, let's try to keep this list to restaurants that have more than a veggie burger and salads. Technically, yes, those menu items are vegetarian friendly, but there is a void in the Triangle for a TRUE list of restaurants with more than a couple of vegetarian items on their menu. It was created for the non-vegetarians out there who have vegetarian friends>>

As usual, it's probably best to check websites for updates to the menu, but this will give you a good start. It also probably goes without saying that most Mexican, Chinese, Indian and Pizza places have vegetarian options.

Raleigh

Restaurant Location Vegetarian Menu Options
SPIZE CAFE Downtown Raleigh Numerous veg options: sandwiches, salads, rice plates, curry and rolls utilizing fresh herbs and Asian spices 
Remedy Diner Downtown Raleigh 12 entrees ("Alternative Therapies"), plus salads & appetizers
Sunflowers Downtown Raleigh

13 lunch items, 1 dinner entree with salads

Irregardless Cafe Downtown Raleigh 5 sandwich items, salads, 2 starters, 9 dinner entrees
Sunflowers Near NCSU 4 menu items, appetizers
Village Deli Cameron Village  
Market Restaurant Downtown Raleigh 3 appetizers, salads, 5 entrees (very seasonal menu, check often)
Rockford Glenwood South Appetizers, 6 sandwiches
Thaiphoon Bistro Glenwood South 2 appetizers, 10 entrees
Sosta Cafe Downtown Raleigh 5 sandwich options
Sweet Tomatoes Capital Blvd Buffet
The Borough Downtown Raleigh Appetizers and 4 sandwiches
Neomonde West Raleigh, North Raleigh, Morrisville Multiple options.  Lebanese.
Humble Pie Downtown Raleigh Tapas, 8 small plate options
Sitti Downtown Raleigh  
Sawasdee North Raleigh Thai base - can add tofu to many dishes
Capital Club 16 Downtown Raleigh Lunch: 3 sandwiches, Dinner: starters, 3 entrees
Sadlack's Heroes Hillsborough St  
Dalat Oriental Restaurant 2109-110 Avent Ferry Road
Mission Valley Shopping Center (Upper Level)
(Corner of Western Blvd. And Avent Ferry Rd.)
 

http://www.dalatorientalrestaurant.com/

Any of their meals can be ordered with or without meat or tofu.

Zest Cafe  North Raleigh

Vegetarian appetizers, seasonal entrees, soup of the day.

The Fiction Kitchen Downtown Raleigh

http://thefictionkitchen.com/

100% vegetarian, including vegan and gluten free options.

Cary

Restaurant Location Menu Options
Udupi East Chatham 100% vegetarian
Cool Breeze East Chatham 100% vegetarian - Indian street food

Morrisville

Restaurant Location Menu Options
Tower Indian Restaurant Morrisville Square

100% vegetarian

Sai Krishna Bhavan Morrisville Square 100% vegetarian

Durham

Restaurant Location Menu Options
Blue Corn Cafe 9th St Handful of vegetarian dishes that can be converted to vegan upon request
Elmo's Diner 9th St  
Toast W Main St

Several crostini and sandwiches are vegetarian.  

Check the specials board for the vegetarian soup of the day.

Chapel Hill

Restaurant Location Menu Options
Sage Timberline 100% vegetarian
Butternut Squash University Square 100% vegetarian

Carrboro

Restaurant Location Menu Options
The Spotted Dog Weaver & Main "Focus on vegetarian" according to their website, and can convert most vegetarian items to vegan.