The House of Lords has been defanged (with several rounds of reduced powers), denying it a true veto of legislation supported by the House of Commons. It isn't doing much active harm and its economic cost has been trimmed.
The proportion of the Lords who are life lords or hold lordships ex officio as a result of some clerical or legal office, has grown, and those are less problematic. In some ways they are more akin to the U.S. federal judiciary or the formal role that the government of France provides to ex-Presidents in its system.
The House of Lords historically was dominated by the Tories and continues to have a plurality of them, so a major political party finds it in their partisan interest not to abolish it. Also, any attempt to abolish the House of Lords brought by another major party can be tainted as appearing to be a partisan power grab. The Prime Minister can appoint new life lords to rebalance the House of Lords and it is less extremely biased on a partisan basis than it once was, but the norms regarding the acceptable partisan mix of the House of Lords as a result of life lord appointments are unsettled and in flux. "Packing" the House of Lords in one session is disfavored, but "rebalancing" it is considered acceptable to some extent.
The abolition of the House of Lords would be a major step towards the disestablishment of the Anglican Church which many conservatives disfavor.
Functionally, as a "house of revision" the House of Lords catches inadvertent errors in legislation that the hastier U.S. style legislative process with long bills finally adopted in the dark of the night with little notice does not allow.
On some rare occasions, the House of Lords has called out politicians in the House of Commons for enacting legislation that impairs human rights, sometimes securing changes as a result.
While the House of Lords collectively, may add only modest value in the decision-making and voting aspect of legislation, the legislative process also benefits from having diverse voices present in the deliberative process that leads to the adoption of legislation, and the House of Lords provides dozens of avenues other than through partisan elected politicians for new voices to be added to that discussion. Even if only 1-5% of House of Lords members at any given time are making meaningful and useful contributions to the deliberative process that are being provided from no other source, those powerful ideas could be beneficial in some critical respect at some point.
Prior to the creation of the U.K. Supreme Court, the House of Lords was the pinnacle of the U.K. judicial system (and that of many former Commonwealth colonies in the form of the Privy Council) and that function created value. This is a recent event and the U.K. may want to have a fallback solution to return to if this major constitutional experiment fails to work as planned.
It provides a way to assuage the egos of the residual aristocracy of the U.K. at a minor fiscal and policy cost since it is only a house of delay. Abolishing the aristocracy entirely could have led to an anti-democratic revolution and postponed democracy in the U.K. considerably. Preserving it allowed the U.K. to get democracy sooner and with less resistance at the outside, while allowing the power of the lords to be gradually reduced over time.
More generally, legitimacy is path dependent. Generally speaking, if something has always been so, and you learned that this was the way that the system worked as a child and came to accept it, and if there is not a break in succession from one political system or regime or set of leaders to another, it will be accepted as legitimate going forward. The House of Lords has met that test without doing great harm.
One can also see the House of Lords as a set of "training wheels" for the democratic government in the House of Commons, because its continuous sessions means that there are legislative institutions (as well as the monarchy) in place during interim periods when a new parliament has been elected but not yet organized a government. While this is probably unnecessary now, and while it isn't obvious what the House of Lords would actually do if the House of Commons failed to reorganize properly following an election, the appearance that there are institutions that aren't disrupted by the election projects an image of stability following elections, as opposed to allowing opportunists to see a momentary power vacuum as the window of opportunity for radical or extralegal change. In this regard, the House of Lords serves of role similar to the largely symbolic legitimacy and continuity affirming process of having Congress count the electoral votes in a U.S. Presidential election.
British political and constitutional history is marked and distinguished by avoiding all or nothing approaches. For example, it retained a constitutional monarchy rather than becoming a Republic (after a brief attempt at that left a bad taste in the British collective consciousness in the 17th century, and then in the 18th century with the American Revolution). Rather than granting full independence to its territories and dependencies, the U.K. has reached individual deals with incomplete autonomy for them. Some have their own domestic laws but remain British for foreign affairs and military purposes (the British demonstrated they were willing to go to great length to honor their obligations in the Falklands War), in some like New Zealand, Australia and for the first century, Canada, it retained final judicial review over its courts, and all have kept a Governor-General's position as a representative of the monarchy. It has different arrangements with Scotland, with Northern Ireland, and with Wales. The House of Lords reflects again the British political culture's tolerance for a situation that is neither an absolute republic, nor an absolute oligarchy or monarchy. France abolished the aristocracy with revolutionary consequences. The British have merely whittled away at it, reducing its privileges, rather than absolutely adopting the principle that all persons are created equal and should be equal under the law for all purposes. As the Daily Express newspaper noted in a January 22, 2020 article:
Desire to do away with the House of Lords has long existed in British political consciousness and was once accomplished nearly 400 years ago.
When Oliver Cromwell came to rule England after the Civil War of the 1640s, he dramatically reduced the House of Lords’ power, and ultimately abolished the upper chamber in 1649.
Parliament branded the Lords “useless and dangerous to the people of England” as they abolished it, but it eventually returned with a resurrected monarchy in 1657.
Afterwards, the Lords briefly enjoyed an increase in their power, able to vote down legislation until the Parliament Act 1911 removed this right.
("Briefly" in this case, meant 254 years, reflecting the British sense of history.)
The preservation of the House of Lords is tied into the norm that British legislators protect the country's constitutional traditions that are only entrenched in ordinary legislation, such as the requirement to hold elections at least every five years and a constitutional tradition of protecting civil and human and property rights. While the institution of the House of Lords adds little value in and of itself, the norm of protecting constitutional traditions that protected it has great value and indeed, is really indispensable to, the U.K.'s democracy since it relies on such norms more than almost any other modern democracy. Abolishing the House of Lords could weaken more important aspects of the U.K.'s constitutional order that lack formalistic protections of the kind found in the U.S. Constitution and the constitutions of most other modern democracies and instead rely on this political norm.
Since aristocratic positions are legally a form of inheritable property, abolishing the aristocracy, or at least seeming to do so by abolishing its most visible vestige in the House of Lords, could also be perceived as an attack on the inheritable property rights that share of common legal heritage with aristocratic hereditary titles and the perception that one is undermining property rights by doing so could also be politically problematic since protecting property rights is another important quasi-constitutional political norm in British politics.
Closely related is that almost any argument for abolishing the House of Lords is simultaneously an argument for abolishing the monarchy, and the monarchy is seen as a valuable institution in the U.K. system as a symbol of the state that is not tied up in partisanship, and as a safeguard in the event of an event that might threaten the existence or legitimacy of the nation such as a dispute over succession to the Prime Ministership or a disaster or military event that disrupts the ordinary operations of government. Constitutional monarchies, in general, have done better at surviving crises than pure Republics. Abolishing the monarchy without broad support could precipitate a constitutional crisis and crisis of legitimacy in the U.K., and abolishing the House of Lords could threaten to hasten such a crisis.
In the current statutory arrangement, the House of Lords is the primary institutional barrier stopping the House of Commons from legislating to cancel the next election and just ruling in perpetuity with no elections. This is because there is an exception to the Parliament Act allowing the Lords to permanently veto any law that could postpone a general election. Arguably, the institutional barrier could be overcome by legislation, just as the House of Commons, in general, has broad authority to reconfigure the House of Lords in general, in a manner reminiscent of the use of the "nuclear option" in the U.S. Senate to abolish the filibuster with only the support of a bare majority of Senators, rather than the two-thirds majority normally needed to change the rules of an institution with continuity of members from one two year "Congress" to the next. But it is a significant speed bump that any effort to do so would have to overcome that could lead to bipartisan opposition that would prevent such a bill from passing in the House of Commons.