Beans, Carrots, Beets
Companion Planting is the placement of various crops in close physical proximity to one another so as to symbiotically compliment each others health, vigor, growth and the flavor of their produce. It also naturally involves separating plants whose development is antagonistic to each other.
Good Companions for Lettuce are beets, broccoli, bush and pole beans, carrots, cucumbers, dill, onion, radish and strawberries.
Avoid cabbage. Cabbage impairs the growth and quality of lettuce. Avoid foxglove. Foxglove aphid attacks lettuce and is often found in joint infestations on either plant - 
Beets are prized for adding minerals to the soil. Leafy veggies such as lettuce benefit from the added magnesium provided by beets and there are no drawbacks for the beets.
"Beets grow well near ....Lettuce and most members of the cabbage family are "friendly" to them. " Carrots Love Tomatoes - L. Riotte
Beans and Legumes
Beans - Both bush and pole beans as well as peas and related legumes grow well with lettuce. In one study Okra and Lettuce + Squash and lettuce planted in conjunction with beans produced yields of 45 -66% higher than those of beans planted by themselves. 
Exceptions to this are with crops susceptible to field history of white mold [sclerotia] a plant fungus. Beans should not be preceded by tomatoes, potatoes, lettuce, crucifer crops, or bean (including soybean). To reduce fungal sclerotia - white mold. Snap Beans being highest on list of beans to be avoided in rotation with lettuce. See- Production guide for Organic Snap Beans  -
Broccoli - Lettuce and broccoli go well together and this combination enables you to achieve a more diverse garden within a very finite space.
Brocolli and Lettuce can be planted relatively close together, even inter-planted without fear of competition for nutrients and space. Their root systems grow at different depths and draw nutrients from different levels of the soil.
Basil - it is claimed enhances the flavor and growth of garden crops, especially tomatoes and lettuce. This may or may not be an Old wives tale, as there is no proof of this.
Carrots grow much deeper into the soil than lettuce. They draw moisture and nutrient from varying levels are are not in competition. Lettuce matures more rapidly than carrots, allowing you to harvest amounts on a daily basis. Planting a row of lettuce between every other row of carrots they work well together and their foliage will add an aesthetic appeal. Using various varieties of lettuce will also add to the effect.
The lettuce serves a semi- living mulch and helps retain soil moisture. Planting them together benefits both.
"Lettuce and carrots are examples of vegetables that can be combined due to biological, nutritional, economic and social reasons, representing an alternative for food production and income." Effect of intercropping on the population density of pests in some vegetables
"Lettuce grows well with ....carrots, and it has long been considered good to team with radishes." Carrots Love Tomatoes: Secrets of Companion Planting for Successful Gardening
Celery also grows well with lettuce for the same reason as Carrots . Celery is a close relative to Carrots.
Parsley, another Carrot family Plant - Ditto.
Lettuce is a cool weather crop while cucumber is a warmer weather crop. Many of their growing requirements differ. The two are not ideal candidates for companion planting but it can be done successfully and somewhat symbiotically as they have no true antagonistic interactions.
Cucumber when it is trellised can grow next to lettuce plants. When properly configured, the vines will provide shade for the lettuce helping them survive the dog days of summer.
Companion planting dill will help cabbage family plants, asparagus,onions and lettuce. Combining beans in a menage a trois of beans dill and lettuce is a scenario I've used with great success on many occasions. The beans fertilize, the dill protects, and the lettuce mulches. Dill repels many pests which feed on lettuce - Cabbage loopers ~ Cabbageworms, and Spider Mites are a few.
Onions repel or deter insect pests that favor leafy crops such as lettuce. Onions and other alliums emit a scent that acts as a repellent. Onions also mask the odor of other crops, confusing pests and keeping them uninterested. Onions will also keep rabbits away.
Onions impede the growth of beans, peas and legumes. Beans and related crops that are compatible with lettuce should not be mixed with onions. Same for Cabbage family plants keep them separated from Onions.
Lettuce is believed to enhance the flavor of radish in the same fashion that chives, chervil, leeks and coriander do. Icicle radishes or other aromatic radishes will repel Cabbage maggot, Squash bug and Squash vine borer.
Strawberries can be intermingled with lettuce very successfully. Alliums such as chives and garlic are sometimes added to this scenario as well. Although there are no documented benefits relative to fertilization, pest suppression, flavor enhancement relative to the strawberries and lettuce, there are also no negative inter-actions.
It is sometimes used for space management and aesthetics.The strawberries can provide a ground cover /living mulch while the mature lettuce adds a protective shelter. Adding alliums to this scenario creates some insect protection.
1. - Flower Flies and other Biological Control Agents for Aphids in Vegetable Crops
2. EFFECT OF INTERCROPPING ON THE PERFORMANCE OF SOME SUMMER VEGETABLE CROPS GROWN UNDER DIFFERENT ROW ARRANGEMENTS
3. Production Guide for Organic Snap Beans