I would go with 'nesting place' - this is commonly used in the UK.

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You're right, or this is what it's called by the RSPCA (Royal Society for the Protection of Animals).

Starlings and tits may also use holes
in trees or other cavities for more protection and to reduce
the risk of eggs or young falling out. Puffins even use
disused rabbit burrows on cliff-top nesting sites.

The space under the roof and eaves were the
most commonly reported nest-site locations.

http://www.rspca.org.uk/ImageLocator/LocateAsset?asset=docum...

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Note added at 12 mins (2013-12-23 22:54:25 GMT)
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It is the most natural of instincts for birds to explore other territories to find a suitable nest site. Even if you have been feeding them all winter, come the spring they will want to pair up and find a place to share a nest. If your garden has all the requirements, they are very likely to stay in the territory they know.

http://www.britishbirdfood.co.uk/help-and-adivce/nest-box-si...

The yearly reproductive cycle begins at the start of the Antarctic winter, in March and April, when all mature Emperor Penguins travel to colonial nesting areas, often walking 50 to 120 km (31 to 75 mi) inland from the edge of the pack ice.

http://en.wikipedia.org/wiki/Emperor_Penguin#Courtship_and_b...

In most mound-building species, males do most or all of the nest construction and maintenance. Using his strong legs and feet, the male scrapes together material from the area around his chosen nest site, gradually building a conical or bell-shaped pile.

http://en.wikipedia.org/wiki/Bird_nest#Mound

http://books.google.co.uk/books?id=BfjUW8ZVinkC&pg=PA57&lpg=...

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Note added at 15 mins (2013-12-23 22:57:08 GMT)
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Sorry, I had forgotten what RSPCA stood for. As could be expected, I had got it wrong.

We're the Royal Society for the Prevention of Cruelty to Animals (RSPCA) and we've been here, looking out for animals, since 1824.

http://www.rspca.org.uk/in-action/aboutus

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Note added at 18 mins (2013-12-23 23:00:05 GMT)
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Where possible, the birds excavate a nesting burrow into the soil. Sometimes they will make use of Manx shearwater or rabbit burrows. Where burrowing is not possible, the birds nest under boulders or in cracks and cavities in cliffs.

The birds defend the nesting site and its immediate surround, and use it in subsequent years. Puffins lay only a single egg, in late April or early May. Both parents incubate it for 36-45 days, and they share the feeding duties until the chick is ready to fledge.

http://www.rspb.org.uk/wildlife/birdguide/name/p/puffin/nest...

I've provided examples that refer to several species.

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Note added at 15 hrs (2013-12-24 13:45:27 GMT)
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FINDING A PLACE TO BREED

Throughout the year, most birds use day length to tell what season it is. When the number of hours of daylight exceeds a certain critical level, physiological changes are triggered in birds which prepare them to breed. Most birds, especially those in temperate regions, also time their breeding activities so that they will be feeding their nestlings when food is most abundant. However, well before nestlings arrive, birds need to select a breeding territory. Non-migratory species may either maintain a territory throughout the winter or establish a new one in the spring. Migratory birds begin looking for and defending a territory as soon as they arrive in the spring. Good territories provide potential nest sites, reliable food sources, and protection from predators.

NEST BUILDING

Nests provide a safe place for eggs and young birds to develop. Bird nests are extremely diverse, although each species typically has a characteristic nest style. Some birds do not make nests at all and instead lay their eggs in a simple scrape in the ground. Other birds construct nests from natural materials, such as grass, leaves, mud, lichen, and fur, or from man-made materials like paper, plastic, and yarn. Nests can be found almost anywhere – on the ground, in trees, in burrows, on the sides of cliffs, in and on man-made structures, etc. Females typically build nests, but sometimes both parents or just the male will build it.

http://nestwatch.org/learn/general-bird-nest-info/nesting-cy...

6. DO ALL BIRDS BUILD NESTS?
Not all bird build nests. Andean Condors lay their eggs on the sandy or gravel surface of a natural cave high in the mountains. Other species will modify a nest site only slightly by scraping the surface of the ground or adding a few sticks or leaves around the eggs. These minimal nest types are most common where predation is not a factor, such as on islands or inaccessible cliff faces.

http://nationalzoo.si.edu/Animals/Birds/faq.cfm

Nesting
Nest site fidelity.
Studies have shown that individuals of most penguin species return each year to the same rookeries. In addition, most penguins return to the same nesting site within the rookery. Studies have indicated that fidelity to the previous year's nest site was 99% for male Adélie penguins, 94% for chinstraps, and 63% for gentoos.
Males arrive first to the rookeries to establish and defend their nesting sites. In a study on Adélie and chinstrap penguins, females arrived one day and five days after the males, respectively.

http://seaworld.org/en/Animal-Info/Animal-InfoBooks/Penguin/...

No need to say "place". The purpose of the nest is to provide refuge, warmth and safety for the eggs to hatch and the birds to be fed until they learn to fly solo.

DLyons is right, not all birds build nests, but we are talking about those that do, otherwise we cannot answer this query.