Names 「名前, Namae」 are one of the fundamental properties of the Tensei Shitara Slime Datta Ken universe alongside Titles and Divine Protections. Similar to Titles, Naming works differently for Humans and non-humans.
At first Naming is a process introduced as only being able to establish master-servant relationships. However, later on, it is shown that Naming is actually much more versatile than it seems, where individuals of equal status name each other, individuals self-name, and individuals name others' subordinates. In other words, as long as proper conditions are set beforehand, Naming can be pretty handy in the long run.
Throughout the story, Human Naming seems to have no intrinsic value, similar to Human Titles, other than recognizing them as individuals with their own identity. Because of how their civilization and society works, Humans are generally named right after they're born, either by their parents or someone related to them.
The story only explains that Monsters are generally unnamed and naming them raises their status. However, it is unknown if this rule also applies to Humans as well, since Humans have been naming each other in accordance to their societal rules, it's possible but still unlikely that an unnamed Human receiving a name from a higher rank being will be given the same treatment as monsters being named. If so, it would explain why the whole of human populace generally remains weak, since weak people naming weak people will only result in weak people.
Names are granted to unnamed monsters by an individual with high magical abilities. By gaining a Name, the monster becomes a Named Monster and obtains a higher status, becomes much stronger than before, even evolving at times.
The stronger the Namer, the bigger the jump in evolution of the Named, and the stronger the Named, the more magicules will be pulled from the Namer to evolve.
For example, when Rimuru named the Goblins, Rigur explained that although his elder brother had also received a name, his evolution wasn't nearly as far as Rimuru's naming. Additionally, after Rimuru named the Ogres, Souei explained that six Kijin appearing at the same time is an unprecedented event, never before seen, since Kijin is an evolution that only the most powerful Ogre achieve over the course of hundreds to even thousands of years.
Moreover, since Ogres are much higher beings compared to Goblins, it only took six of them to make Rimuru enter his 'sleep mode' compared to when he named 100 Goblins plus Ranga.
Since naming a monster requires magical energy, most individuals refrain from doing so, as most of the time, the magic energy lost due to naming will stay permanently lost leading to a permanent decrease in the Namer's total magic capacity. Sometimes, it is even possible to outright die due to excessive loss of internal magicules. However, Rimuru managed to luckily stay alive by recovering his lost energy from Veldora who lay sealed inside him as backup, while in his 'sleep mode'.
Thus, since naming is such a rare occurrence among the monster community, almost all of which end up living their entire lives without names, any monster becomes amazingly grateful to the one that have named them, revering that individual in the likes of a King or even their God.
If the master dies, a monster’s named status will disappear as well. This was why Rimuru was able to rename Gabil as Gabil again. It was assumed to be a name overwrite, but it was simply that Gabil no longer had an authority to protect his identity.
If a Demon introduces itself with its real name, then that name can be used to manipulate it. Therefore, not introducing oneself was common sense among demon-kin.
There are many individuals seem among the Demi-Human community who have Names but don't seem to have that big of a change, such as Kaijin and co., the Elves in the Dwargon bar, Treyni and many others. It is likely that they follow the same Naming scheme as Humans, and thus have no significant effect other than individualism.
However, there are others, such as Ramiris, Dino, Dagrule, Ruminas and the four True Dragons who seem to have names without anyone naming them. Although it is possible that Veldanava named the former three and his siblings, it doesn't explain why he didn't name Guy as well even after he gave him such a big role similar to Ramiris and Dino. Moreover, it also leaves the question of who named Veldanava.
Since it isn't possible for someone to name Veldanava, the only other explanation is that he named himself and that there also exists a system where individuals can choose to 'self-name'. Such a phenomenon would probably also exhibit properties similar to that of Human naming.
Equal Status Naming
The very first occurrence of naming introduced in the story was actually equal status naming, when Rimuru established the name 'Tempest' as his and Veldora's surname. When two individuals share the same last name, it establishes their position as equals and not as master and servant. This can have a heavy impact on the future growth of both individuals as equal status naming means that both parties involved have a higher degree of freedom when exchanging energy and using each other's abilities. Unless the other party outright rejects a proposal, the former party is capable of using the full arsenal of the latter.
An example of this would be the first time use of the Veldora Sword, where Rimuru lightly flicked his wrist, but the power output of the sword was still monstrously high. Since Rimuru hadn't explicitly told Veldora to keep the output low at the time, Veldora put too much 'oomph' into it.
In other words, if Veldora and Rimuru had not gotten their last names, Rimuru would've been recognized as Veldora's subordinate, and Rimuru's growth would probably not be as great, since a master-servant relationship basically always keeps the servant as a servant no matter how fast or great they grow. Ultimately, Rimuru being recognized as equal to Veldora basically meant that he was recognized as a being on the same status as a True Dragon, and because of this, it was required for Rimuru to actually become as strong as a True Dragon to live up to it. Hence his growth.
Naming Others' Subordinates
Around the middle of the story, the flexibility of naming is truly shown. As Ramiris starts living in Tempest, and becomes manager of the Labyrinth City, Rimuru and she had shared ruling over some individuals, namely, Beretta, Zephyrus, Boreas, Notos and Euros. Originally, Beretta was summoned to be ruled under Rimuru after the 100 year protection contract with Ramiris. But since Ramiris saw that Rimuru already had way too many powerful subordinates, she nagged him into permanently giving Beretta to her, and since Beretta also took a liking to the cheerful fairy, Rimuru was willing to transfer over the primary master to be Ramiris while he himself acted as the secondary. Thus, even if Rimuru isn't the true master, as long as he has a soul corridor established, he can name and evolve almost anybody he wishes.
Alternatives to Magicules
Naming requires an enormous amount of Magicules, simply that. In other words, as long as the Namer had enough magical energy to supply, the naming can be done. This means that it could be any source of magical energy, as long as s/he has direct access to it. An example of this would be when Rimuru used the energy produced after disassembling 5,000 souls each, from his Stomach, to name the four Dragon Kings.
Almost everyone in Tempest is a named monster.