Looking through a 3.5e adventure it mentions a Diplomacy roll:
Give the PCs a minute to explain themselves to the lich, at the end of which one of them can make a Diplomacy check in an attempt to improve his hostile attitude.
But I am running it in 5e. The closest check is Persuasion which doesn't feel quite the same.
How do I run this part of the adventure in 5e?
First, I want to point out a bit of a paradigm shift from 3.5 to 5e: Players attempt things and then the DM requests the "appropriate" skill (5e), rather than the players attempting to use a skill (3.5).
Keep in mind, the action your player describes/requests may not fall neatly within the Charisma(Persuasion) paradigm. For the sake of explanation, we'll assume that it does and then elaborate at the end.
PHB p174 reminds us:
The DM calls for an ability check when a character or monster attempts an action (other than an attack) that has a chance of failure. When the outcome is uncertain, the dice determine the results.
For every ability check, the DM decides which of the six abilities is relevant to the task at hand and the difficulty of the task, represented by a Difficulty Class.
And then we go down to each ability (p178):
A Charisma check might arise when you try to influence or entertain others, when you try to make an impression or tell a convincing lie, or when you are navigating a tricky social situation. The Deception, Intimidation, Performance, and Persuasion skills reflect aptitude in certain kinds of Charisma checks.[snip]
Persuasion. When you attempt to influence someone or a group of people with tact, social graces, or good nature, the DM might ask you to make a Charisma (Persuasion) check. Typically, you use persuasion when acting in good faith, to foster friendships, make cordial requests, or exhibit proper etiquette. Examples of persuading others include convincing a chamberlain to let your party see the king, negotiating peace between warring tribes, or inspiring a crowd of townsfolk.
If the player describes a well constructed and/or heartfelt explanation as to why the party deserves leniency, then that's going to fall under my emphasis on "Negotiating Peace" and Charisma(Persuasion) is going to apply.
This is further supported by the fact that various opportunities to "convince" NPCs in adventures use Persuasion, Deception, or Intimidation (I looked through the Waterdeep Dragon Heist adventure).
If the party/member is trying to lie or bluff in the traditional sense, that would be a Charisma(Deception) ability check.
Furthermore, in 5e, the DM has a bit more power to choose these sorts of things: You could decide that it's a miscellaneous Charisma check (with no specialized skill) or even an unusual combination of Ability and Skill (PHB p175), such as Intelligence(Persuasion) to represent remembering legends and lore about how to appeal to Liches.
The important thing is to let the party's action dictate any rolls, not the other way around.
Generally speaking, 5e Skill DCs have a lower cap than in 3.5. This is due to bounded accuracy; The highest reasonable bonus on a skill is around 17, barring any shenanigans, compared to easily getting over 30 in 3.5 from your ability score, skill ranks, and magic items..
\begin{array}{lll} \rlap{\textbf{Example DifficultiesClasses}} \\ \textbf{Difficulty} & \textbf{3.5 example} & \textbf{5e example} \\ \hline \text{Very Easy} & \text{0} & \text{5} \\ \text{Easy} & \text{0} & \text{10} \\ \text{Average\Medium} & \text{10} & \text{15} \\ \text{Tough\Hard} & \text{15} & \text{20} \\ \text{Challenging\Very Hard} & \text{20} & \text{25} \\ \text{Formidable} & \text{25} & \text{---} \\ \text{Heroic} & \text{30} & \text{---} \\ \text{Nearly Impossible} & \text{40} & \text{30} \\ \end{array}
