"An influential member of parliament has not only to pay much money to become such, and to give time and labour, he has also to sacrifice his mind too - at least all the characteristic parts of it which is original and most his own."
Walter Bagehot
Many atheists and secularists do not think there should be bishops in the House of Lords; the idea that these "Lords Spiritual" get to vote on legislation which everyone then has to follow seems obscene. We are a secular nation, more or less, yet 26 Church of England Bishops get to sit there being all unelected and acting like they have some kind of revealed wisdom. This doesn't just seem odd from the perspective of a secularist, a Quake or a follower of Islam; it is also at odds with the view of UK Christians in general of whom only 12% believe that religion should have special influence in public life.
I disagree with this view of Bishops in the HoL, provided there are several hefty strings attached. Here is an over-simplified summary of what is required for legislation to become law; it must be drafted and passed in the House of Commons ("The First Chamber"), it must then be passed in the HoL ("The Second Chamber"); the HoL can suggest amendments and send it back to the HoC for these to be passed.
There is are plenty of problems with the HoC, two of which are important here; firstly, members of the HoC are forced to balance the long-term interests of the country with the interests of their voters and this means they will sometimes do the wrong think to please their voters (in the words of the great scholar Super Hans, "People like Coldplay and voted for the Nazis. You can't trust people, Jeremy"); secondly, the members of the HoC are almost all professional politicians, whilst quite a few were once lawyers there is a shameful lack of scientists, engineers, healthcare professionals and other experts with the end result being that the HoC doesn't always know what's best for the country.
For the HoL to add anything to the system, it needs to combat these problems; currently, the Lords do not have any voters to please and they are not in real danger of being removed from their role allowing them to act for the long-term benefit of the country; additionally, many peers gain their role because they are specialists in a particular area and this means they can scrutinise bills that pass through.
Making the Lords elected would make the HoL a valueless copy of the HoC (there is the suggestion that a longer time between their elections would allow Lords to act more in the country's long term interest however that just seems to water down the previous problems) so here is my suggestion, we create a HoL that is dynamic, specialised and diverse. Currently, not all Lords vote on all issues, many Lords don't even show up so instead of allowing any Lord to sit and vote on any issue, we tag each Lord with areas of expertise in which they can speak, vote and otherwise pontificate.
With this more dynamic, "mix-&-match" HoL we could increase their numbers allowing greater diversity and include individuals with expertise as well as individuals from various walks of life. By setting topics for Lords before they are appointed we can avoid anyone selecting favourable Lords to vote on a chosen issue; for those occasions where a Lord has expertise that isn't specified, the other Lords for that topic could vote to allow them to speak.
It would be no biggie to include lawyers to ensure law is written in a clear and unambiguous manner following this we can add scientists and statisticians to check for factual accuracy; philosophers (or logicians?) to check for logical consistency; even academics of social or political science to criticise the legislation. After all this scrutiny, the bill has to be approved by the HoC anyway so the democratic safeguard against some kind of totalitarian elitism is maintained.
Where do Bishops come into this? Whilst I don't think they have anything that could meaningfully be called expertise, they do represent the views of a meaningful number of citizens and therefore come under the various walks of life category along with Imams, Rabbis and (I'm being serious here) Jedi Masters. In this new HoL, religious representation is not that big of a deal so it gets to stay.