Hi Jono
Brünnich's Guillemot may prove to be difficult in mid-August I'm afraid Jono. They tend to have left the cliffs by the first week of August (I don't think I've ever seen them after 10th August) although there is always a chance that you could see them on the sea below the bird cliffs.
For a one day trip with the family then the obvious choice is round the Snæfellsnes peninsula (which is the long narrow peninsula jutting out from central western Iceland). Although birding activity is past its best in August, there are still plenty of things worth seeing and the scenery of mountains, waterfalls, bird cliffs, lava fields and a whopping great glacier-encrusted volcano is more than enough to keep the family entertained. It's doable in a day, a longish day, but you won't regret it, unless it's howling a gale and the rain is horizontal.
There are auk colonies at the cliffs called Þufubjarg between Hellnar and Malarrif (these are collections of houses rather than villages) and also at the very end of the peninsula at cliffs known as Svörtuloft. As I say the auks will have left the cliffs but may be loafing around on the sea below.
Also worth checking out are the cliffs at Arnarstapi as you can usually see Harlequins at this time of year on the rocks at the base of the cliffs and it's all very picturesque (no Brünnich's here though).
There is a whole series of lakes along the southern side of the peninsula which hold wildfowl, Great Northern and Red-throated Divers and Slavonian Grebes.
The northern side of the peninsula is also extremely scenic. The lake at Rif usually has hundreds of Red-necked Phalaropes in late summer although they may have moved on in mid-August. Check for Grey Phalaropes and possible Nearctic waders - I've seen Semipalmated Sandpiper and Pectoral Sandpiper here in the past.
Also plenty of Glaucous Gulls breeding in this area, especially closer to Grundarfjörður.
It's generally not a good area for Gyr Falcon in summer. I have seen them here but only rarely. NW and NE Iceland are your best bets in summer, although they can turn up anywhere.
As for in Reykjavík itself, then you can stroll through the Fossvogur cemetery which is just south of the prominent building the Pearl (Perlan) and try to get to grips with Icelandic Redpoll ID and see some extremely tame Redwings. Better still is to go to the lake Bakkatjörn which is in the suburb of Seltjarnarnes at the very end of the little peninsula where Reykjavik is located. Apart from Icelandic birds, it is a regular vagrant trap and has had Western Sandpiper in the past (I wouldn't bank on it this time), Sabine's and Bonaparte's Gulls in summer and at least four different Ring-billed Gulls this year.
The bay immediately south of the aforementioned cemetery (Fossvogur) is also OK for passage waders at low tide. If you see someone birding by bike there, it's probably me as this is my local patch.
Puffins are probably best seen locally on special boat trips from Reykjavik harbour or from one of the whalewatching boats in Reykjavik. Whalewatching trips from Reykjavik last August sometimes yielded 40-50 Sooty Shearwaters.
E