No, it's a hereditary position, although there was a bit of push and shove about it a few centuries ago. If the royal family could be selected the whole thing would have vanished eons ago. The British Parliament and local government is elected.
There were occasions when the reigning monarch died without an heir, Queen Elizabeth 1 and Queen Anne but their heirs were chosen from among relatives. For instance Queen Elizabeth was succeeded by James 1, he was the son of her cousin, Mary, Queen of Scots but he could also trace his lineage back along the lines as Elizabeth.
i was taught that the royal charter says that the subjects have the right to appoint the royal house. the current royal line is illegitamatly descended anyway, from about 1450 ish as i remember.
As I understand it, Britain has a constiutional monarchy, the definition of which I found here which does say either elected or hereditary
http://www.webster-dictionary.org/definition/constitutional% 20monarchy.
But regarding your answer lobster, who are the subjects - is the people or their elected representatives that choose? As pointed out in my own answer there were occasions where a British monarch died without children but their heirs were chosen by Parliament from among distant relatives.
I'm sure I've heard somewhere that English monarchs become the sovereign by popular acclamation, not by coronation. So Edward VIII was king though he was never crowned, and Lady Jane Grey was never queen, though she was crowned. Taking that to its logical conclusion the "heir" to the throne could be refused by the masses - or "them asses" as I have seen it misprinted!
is it true that the queen owns most of washington dc and subliminly rules the american government because of it? the state belonged to british rule way back in history and was never declared american.