Yes, you're right, no one elected Gordon Brown to power... Oh wait, yes they did:
(Results for the 2005 Kirkcaldy and Cowdenbeath parliamentry elections)
I do believe those are the results for electing him to his constituency, not voting for him to be Prime Minister (which is what the GP was alluding to.)
Technically, no UK prime minister is elected by the electorate to be prime minister - it's other members of his party who do that, however the closest to the public voting for them tends to be a general election where the whole of the country votes, of which Mr Brown has yet to endure. He had a chance late 2007 when rumours were rife that one would be held but he/his advisors backed out of that one.
If his party retains a majority in the election in May(? latest it can be held is June 3rd), it could (tentatively) be said that he had been elected by the general populace, but not until then.
Another measure of 'popularity' could be certain No. 10 e-petitions.
One calling for him to resign:
An opposing one calling for him to stay:
Of course, these tend to be self-selecting, but they're just as valid (i.e. not very) than saying that Mr Brown has been elected by the whole of the UK, when the only reason he's there is because he was elected in his constituancy, and a group of his mates who were likewise elected 3-4 years put him there.