The Founder Members of The Football Association 1863

Barnes (London) participated in the first ever match under FA rules, a 0-0 draw against Richmond (who were not FA members). They also competed in the first ever FA Cup and continued to do so through to the 1885–86 competition. Switched to Rugby Football.

Blackheath FC (London) founded in 1858, is the oldest open rugby club in the world. Withdrew from the FA in protest at the proscription of 'hacking', concentrating on Rugby Football.
Blackheath Proprietary School (London) were also in on the foundation of the FA.

Civil Service F.C.  (London)  another club who played both Association and Rugby Football- when the rift over the rules led to other clubs focusing on Rugby, Civil Service continued in both codes. Literally ambassadors of the game, playing fixtures overseas.

Crusaders F.C.  (London) the club continued to play in the FA Cup into the 1880's, but I cannot find any more information about them.

Surbiton (London) - of which I know nothing.
Crystal Palace F.C (London) - not the Crystal Palace who still grace the Football League. This club was formed by groundskeepers at the Great Exhibition site at Hyde Park and lasted from 1861-1876.

Leytonstone Forest (Leytonstone, Essex) - also known as Forest FC,  the ex public schools club that evolved into The Wanderers. 

N.N. Club or N.N. Kilburn (London) -  The No- Names provided the FA with its first president, Arthur Pember.
 Kensington School (London) -  a school that favoured 'dribbling' types of football (like Eton and Charterhouse) over the carrying game (Rugby)
Perceval House (London) a school from Blackheath who also favoured the 'hacking and handling' route. After leaving the FA in December 1863 they vanish from history.