Graham said that MBS (the Crown Prince) "has to go" and and called him "crazy" and "dangerous". It looks like Graham, although a Republican, is strongly against the White House' position on this, which is much more appeasing, e.g. saying that they don't know whether MBS was involved in Khashoggi's murder. Why is Graham coming so strongly against MBS? What's in it for him being the/an anti-MBS champion?
Graham was a member of the Senate Intelligence Committee from 2007-2009 and has numerous contacts within the intelligence services. Likewise, he's a noted hawk on the world stage, supporting an interventionist viewpoint in a party that is increasingly anti-intervention.
It wouldn't surprise me if he were tipped off as to the truth of Turkey's accusation on 10/7 or 10/8, but was unable to comment on details for fear of revealing his source until he could be officially briefed by the Director of the CIA. That's how you see Graham going from full in-Trump's-pocket in regards to the Kavanaugh hearing on 10/4, and then suddenly go off message (and out of Trump's good graces) by 10/8, demanding loudly on TV to be provided details.
MBS is unpredictable, ambitious and a wild card for the establishment
We must first take note that Saudi Arabia has an entirely different "system of values" compared to the West. Public beheadings, whipping, accusations of witchcraft, arrests of dissidents, foreign interventions, funding of various Islamic groups, brutal Sharia laws etc ... This didn't start with MBS, and the Washington establishment (Republic or Democrat) never had many problems with that. Relations between Saudis and US are based on a simple principle: Saudis sell their oil for USD exclusively and attack Israel only rhetorically. But, they are firm opponents of Shia Muslims (Iran in particular). In return, they get protection from the US.
So what about MBS? While being far, far from perfect, MBS is at least doing some steps towards modernization of the country (like female driving for example). In fact, a few years ago he was viewed as very much liberal. What makes him dangerous for the establishment is his unpredictability. The US is now less dependent on foreign oil, but countries like China have increased consumption and would like to trade in other currencies than USD. MSB even had some talks with Russians and Iranians and seems to want to improve relations. This increased independence in foreign relations is not liked by the Washington establishment.
What about Lindsey Graham? He is a precise walking and talking embodiment of the aforementioned establishment. Called RINO and GOP-e, he was and still is despised even by his fellow Republicans, especially due to his "bipartisanship". During the Republican primaries, he opposed Trump and lost badly. After that, he sometimes sided with Trump in Senate, sometimes opposed him. With increased partisanship in US politics and society, lukewarm politicians like Graham have less and less chance of being elected. What he needs is an issue that would make him stand out in the crowd. And the Khashoggi incident is exactly such an issue. Criticizing from a moral perspective, removing one figure (MBS) that is already problematic for the establishment and putting Saudi Arabia back in the US sphere of control while also earning brownie points from the electorate. From his point of view, this is a win-win situation.