Forgive my ignorance of British politics, but isn't Brighton and Hove already represented by the only elected Green in the commons? Or is Lucas representing another constituency in the general area?
She represents Brighton Pavilion, which is one of three constituencies covering Brighton and Hove, and which includes this ward. The other two constituencies, Brighton Kemptown and Hove, are both Tory-held marginals.
The Green council seems to be a disaster and consumed by factional infighting between "mangos" and "watermelons". To add to the food theme the Council leader's surname is Kitcat. Someone who knows the area better may be able to explain more.
The main issue, at least in terms of this by-election, is the strike by the Cityclean (refuse and recycling) workers, and the events surrounding it.
It started with the Green administration (well, the Green Councillors on the Policy and Resources Committee, with the backing of the Tories, and with Labour opposing) voting to hand over negotiations over pay and allowances to the council officers. Whatever their intention, the effect was that many highly contentious potential political decisions were handed over to unelected officials. One group particularly affected by the proposed changes (along with care workers) were refuse and recycling workers, whose pay structure has been seen as archaic by council management for many years. Whatever the details of the proposed reforms, the feeling of the Cityclean workers and their union (the GMB) was that many of them would face a significant reduction in their take home pay.
In the UK generally, refuse workers have the archaic qualities of being manual workers who almost earn a median average wage (shock horror!) and are actively unionised. This isn't a great group for the management of a council - let alone an elected administration - to pick a fight with, as if they withdraw their labour they have an immediate and visible affect in that the city quickly gets covered in rubbish. This is what happened, for a couple of weeks or so (can't remember), until the proposal was more or less reversed, with some caveats. I would argue the Green administration should've seen this coming in the first place, and that it was politically naive of them not to have done so.
Whilst not from the best journalistic source, this is what it looked like:
http://www.heart.co.uk/sussex/news/local/brighton-and-hove-bin-strikeAt the same time, many Councillors, Green Party members, and the local Green MP Caroline Lucas, actively and publicly supported the striking workers, and it is a alleged that some Green Councillors attempted to oust their leader (Cllr Jason Kitcat). This fed into the wide public perception that the local Green administration, and broader membership locally, were somewhat divided.
Hanover and Elm Grove is a city centre ward (which I lived in for two years) where recycling and refuse is already an issue, due to the relatively transient population, large number of houses of multiple occupation, and reliance on communal bins. Although everywhere in the city was affected, this ward would've been particularly badly affected (in fact, wards with Green Councillors would all have been disproportionately worse hit). This is a very recent, and unpleasant memory, and so would've been on the mind of everyone who voted in yesterday's election.
What a mess. The result was exactly what I'd predicted by the way, but anyone can claim that in retrospect.