Today's choice of prime minister by the king was pretty obvious.
But in general: Can I legally challenge that the king appointed X for prime minister? I could e.g. argue that the majority of the MPs do not support him/her, or that he/she was not correctly chosen as head of the political party he/she claims to represent.
Usually, the king has to follow the "advice" of the prime minister for all decisions, but in this case is probably an exception.
There is no direct legal mechanism for an ordinary member of the public to challenge the monarch's appointment of the Prime Minister such that it could force a different appointment.
The Monarch alone appoints a Prime Minister using the royal prerogative.
The Prime Minister holds office until they resign.
The Prime Minister can resign of their own accord.
The House of Commons could vote on a motion that "this House has no confidence in Her Majesty’s Government". It is convention that the Prime Minister resigns when it's clear they do not have the confidence of the House of Commons and that an alternative government (formed from the current composition of the House) does have the confidence.
The ordinary member of the public could write letters, campaign, protest etc to try to influence that.
There's an important step missing from the question (and the other answer): the incoming PM is appointed on the advice - or at least, the recommendation - of the outgoing PM*.
It is assumed that the outgoing PM will make a sensible choice - and they are entitled to wait until a suitable candidate emerges, as happened in 2010.
In that regard, the King is essentially unable to ignore that advice, and convention is that only the Commons can challenge the subsequent appointment, initially by voting down the subsequent King's Speech, or later via a vote of no confidence in the government.
There is precedent for a legal challenge to a PM's advice, as was used during Brexit. However, the circumstances of that challenge were quite specific, and might not apply in the appointment of a PM - especially since the aforementioned mechanisms exist to remove a PM.
* Regarding the role of the outgoing PM on the choice of their successor: it's a matter of convention, but the Cabinet Manual has this to say:
Historically, the Sovereign has made use of reserve powers to dismiss a Prime Minister or to make a personal choice of successor, although this was last used in 1834 and was regarded as having undermined the Sovereign.
In modern times the convention has been that the Sovereign should not be drawn into party politics, and if there is doubt it is the responsibility of those involved in the political process, and in particular the parties represented in Parliament, to seek to determine and communicate clearly to the Sovereign who is best placed to be able to command the confidence of the House of Commons.
As the Crown’s principal adviser this responsibility falls especially on the incumbent Prime Minister, who at the time of his or her resignation may also be asked by the Sovereign for a recommendation on who can best command the confidence of the House of Commons in his or her place.
(Cabinet Manual, 2010, section 2.9)
