It is named after the Greek goddess of memory.
Bugs rose to become captain of her own ship, and assembled a like-minded crew who never stopped believing that Neo was still alive but trapped in the Matrix, even after everyone else gave up hope years ago. They also remained dedicated to freeing more humans from the Matrix, when as the years passed General Niobe wanted to focus more of the resistance's resources on rebuilding their settlements in the real world.
Trusted members of Bugs' crew include field members Lexy and Berg, as well as their Operator Sequoia. The ship's medic is Ellster - who is actually the granddaughter of Captain Roland, one of the few ship captains besides Niobe herself to survive the Battle of Zion sixty years ago. The ship's pilot is Hanno.
As Bugs and her crew believe in Neo's legacy and the Truce he helped make with the Machines, several of its crewmembers are Machines themselves - preferring the term "Synthients" (Synthetic Sentients). They are some of the many Machines who defected to the humans' side after the Machine Civil War broke out. Synthient crewmembers of the Mnemosyne include Cybebe, Lumin8, and Octacles.
A recent addition to the crew is the new "Morpheus" sentient program, who Bugs helped free from the Matrix.
The Mnemosyne was built decades after Zion's hovercraft such as the Nebuchadnezzar, therefore there are numerous design upgrades both large and subtle throughout the ship - even the chairs look sleeker than the ones on the Neb.
The most important aspect of its design is that it was constructed with the help of free Synthients that switched to the humans' side, and as a result it has a hybrid design between old Zion hovercraft and Sentinels. The forward half of the ship has hoverpads like an older human ship, but the rear section ends in a glowing red "tail" that sways back and forth, like the armor on a Sentinel.
Concept art also confirms that the Mnemosyne has a "stealth" mode, aiding it in penetrating into enemy power plants.
Behind the scenes videos clarify how the ship can be newer and more advanced while also having a nameplate stating it was built long ago: the base hull was originally an old Zion hovercraft that was downed at some point and abandoned, but many years later IO found the hulk and salvaged it. They then gutted it and mostly rebuilt it.
The inscription on its core reads "Mark III No. 19 made in the EU year 2074.", according to a plaque shown by Lana Wachowski. However, the final version of the plaque, that ended up in the movie, has the year 2274 on it.
Behind the Scenes
Lana Wachowski revealed the ship's nameplate for a charity auction in December 2020. In the video, Lana said that it continues their "tradition" of having hoverships with difficult to pronounce names (because they're Biblical or philosophical references); she joked that it's "pronounced many ways in the show - sometimes 'Knee-mo-sign', sometimes "Ni-mo-sin-ee'..." - most characters including Bugs herself actually say it as "Nem-oh-seen". 
"Mnemosyne" is the Greek goddess of memory, and from this stems words such as "mnemonic" - something that aids in memory. Keanu Reeves himself played the starring role in the the 1995 film adaptation of the classic William Gibson cyberpunk story Johnny Mnemonic.
The Mnemosyne crew members who use RSI's in the Matrix itself (Bugs, Berg, Lexy) all at various points wear chain-link necklaces, as well as prominent blue clothing (Lexy has a blue jacket, Bugs's hair is dyed blue). It appears that they have essentially "reappropriated" symbols of human slavery in the Matrix, as an act of defiance - comparable to how the pink triangle symbol actually started out as an identifying badge for gay prisoners in Nazi concentration camps, but was then reappropriated as a pride symbol. Their chain necklaces evoke chains of slavery, while "Bluepills" are humans trapped in the Matrix and unaware that they're slaves to the Machines. Bugs and her crew defiantly reappropriated this symbolism by prominently wearing blue colors.
The crew of the Mnemosyne seem to be loose spiritual successors of sorts to E Pluribus Neo (EPN), a radical faction that emerged in Zion during The Matrix Online. Founded by the Kid, EPN was dedicated to upholding the values of Morpheus and Neo.
- Ominously, Mark 3:19 in the Bible is the end of the list of Jesus's twelve Apostles, which reads: "...and Judas Iscariot, who betrayed him." - This might refer to the fact that Bugs defied Niobe's orders to save Neo and Trinity.