I'm going to disagree with AndyWG about Wildwoods and Wholesale Tropicals. Both can easily be reached by public transport, and both are within an "all-zones travelcard". Wholesale Tropicals is about 5-10 minutes ride on a number 8 bus from Liverpool Street station (on the Circle and Central tube lines). Ride the bus towards Bethnal Green, and get off after about a mile, at the stop by a post office (look for the big red/yellow post office sign). Wholesale Tropicals is on the other side of the road.
Wildwoods is about 10 minutes walk from Crews Hill railway station. The quickest way to get there is by Victoria line tube to Finsbury Park, and then go up to the mainline railway and take a commuter train north (the trains going to Hertford North). From Finsbury Park it's about 25 minutes or so, about 10 stops. Not terribly exciting but you'll get to see lots of suburban London, not to mention Alexandra Palace. Once you're at Crews Hill, you go out the station, walk east past all the garden centres (and several other fish shops) and around a corner, for about ten minutes. On the east side of the road is Wildwoods. Wildwoods is definitely worth a visit, but perhaps not for the whole family unless they're really into fish as well!
OK. As for proper London things, here are my three hidden gems:
Sir John Soanes Museum. Utterly bizarre collection of paintings, architectural nick-nacks, fossils, Egyptian stuff, and more. This was an 18th Century collector who was also an important architect. The highlight is the Hogarth gallery, which has moving walls. Free.
http
/www.soane.org/
The Horniman Museum. Old school museum in some ways, but with a modern and very good aquarium. Contains a bit of everything. Originally a private museum, has important collections of things like animals, musical instruments, and ethnic art. Free.
http
/www.horniman.ac.uk/
The Wallace Collection. Originally a private art collection, then left to the state on the proviso nothing was changed. The result is an enchantingly diverse collection of artworks without the fluff typical of modern "interactive" museums. Lots of famous paintings, as well aa a very important collection of arms and armour. Free.
http
/www.wallacecollection.org
All these three places are, for some reason, unfamiliar to the mass of tourists who traipse around the Natural History Museum and such instead. So they are much more fun to visit, and you have the plus of being able to tell your friends back home you did something un-touristy. The Soane Museum in particular is so bizarre and so quaint that every American I've taken there has gushed about it afterwards. It is exactly what you expect an Olde Englishe museum to look like, but more so.
A few other interesting places well worth considering are
Samuel Johnson's house, the
Design Museum, and the
Victoria and Albert Museum. The
Cabinet War Rooms and
Imperial War Museum are also excellent and very interesting for people who enjoy history or militaria. The
Royal Air Force museum in Hendon is another museum off the beaten track but well worth seeing if you like aircraft.
Greenwich is also a good day trip, with some excellent museums and things to see, but also a tourist trap as far as snack bars and gift shops go, so be prepared.
Cheers, Neale