The Government will soon submit a proposal to do just this, with plans to enact this proposal --- if signed into law --- after the next general election.
The House of Lords is a curious institution. Membership had long been largely inherited, although this is being phased out. Today, new members are appointed to life terms, with a substantial number of Anglican Bishops. (There are no seats set aside for any other religious groups.)
The House of Lords largely serves as a counterweight to the House of Commons, although the "lower" house (of the Commons) can overturn (by supermajority vote) the decisions of the "upper" house (of the Lords). Prime Ministers had originally been selected from the House of Lords and, thus, were not popularly elected. Moreover, the House of Lords also includes the Law Lords, whose constitutional place is not unlike US Supreme Court judges except that they also sit in the legislative body, the House of Lords. (Thus, the separation of powers in this and other cases is not clear.)
Whenever I discuss the House of Lords with friends and colleagues in the US, they are almost always surprised to learn that such a legislative body exists. It strikes many that there is so much about the institution that is objectionable. How is it, I am asked, that the UK can ever tolerate an unelected upper house of their Parliament?
I will be disappointed to see all the proposed reforms passed. In my view, one major reason to keep the House of Lords largely as it is composed now is simply because it works. It is perhaps surprising to those outside the UK to learn that the House of Lords has played a crucial and positive role in protecting civil liberties over the last decade especially. The fact that the House of Lords often seems to work so well is that its members can have the long-term interests of the country in mind, rather than newspaper headlines or general elections. Moreover, those that have been appointed represent special expertise across a range of subjects. For example, current Lords include Baroness (Onora) O'Neill, Lord (Bhikhu) Parekh, and Lord (Raymond) Plant. (For details on how to become a Lord, see here.)
A further reason I believe the House of Lords should be retained is because it strikes the correct balance between the elected and the experts. As I have argued before (here) and (here), it is right that those popularly elected can overturn the decisions of the unelected. However, it is also best that governing brings together those with special expertise to help popularly elected politicians rule best.
It is often said that the jury trial is like a mini-Parliament: nothing could be more true. The jury always have the final say on all decisions, but their verdict is informed by the good counsel offered to them by the unelected judge, selected for his or her because of his or her legal expertise. To follow this analogy, the House of Lords is like the wise judge offering its advice when appropriate to the jurors, the House of Commons. The Commons may accept or reject this advice, but the two most often work together in tandem. That this system works is clear from the history of this peculiar institution and the sound decisions it most often reaches, not least post-9/11.
My strong support for the House of Lords is not a support of all features of this institution. For example, I would strongly support not only the end of inherited peerages, but also the end of giving seats to members of the Anglican church only. (Whether it is preferable to distribute the seats to leaders of more religious groups or reserve no seats I put to the side for now.)
All institutions require some reform from time to time. That said, I do hope that the reform of the House of Lords is not as radical as has been proposed and that the distinctive constitutional role it has played in British history and politics preserved for the benefit of us all.
UPDATE: I have posted on more recent developments above here.