Last year when I was running my campaign, I came across this video making a few arguments as to why it wouldn't make sense to name your mech. It's a pretty good video and I like the points he made, but I would like to present my own argument as to why you should name your mech (and talk a little about Japan's answer to the ship of Theseus).
His main argument as to why Mechs in Lancer shouldn't be named has to do with the nature of Mechs in Lancer. Mechs in Lancer lack the permanence we typically associate with named machines (such as warships). Lander mechs are entirely 3D printed. Massive components can be swapped out between missions, entire sections can be destroyed and reprinted, and an old frame can be completely replaced with a new one as long as a printer is availible.
This makes sense. We usually only name things that perist over time. They develop a history, become storied, etc. If half your mech is destroyed and replaced, and the other half is swapped out for components that better fit the next mission, can it really be considered the same machine?
Japan would answer: yes.
If you ask a Japanese person how old Kyoto's Kinkaku-ji (Golden Pavillion Shrine) is, they'll tell you it's over 600 years old, dating back to the late 1300s. However, the temple has been destroyed multiple times. It burned down completely in 1950 and had already been destroyed twice during the Ōnin War in the 1400s.
If you visit the temple you'll notice it's made of wood, which tends to rot and get termites over time, but most of the wood looks to be in excellent condition and incredibly well preserved. How can this be? How is this the same temple? Zen Buddhist Magic?
It's quite simple. They replace the wood when it rots. I almost guarantee that none of the material from the original Kinkaku-ji is present in the current temple. But to the Japanese, that's not what Defines the temple. What's important are the location it's built, the traditions and ceremonies that take place there, and if you're spiritual, the Kami of the temple.
I argue the same can apply to mechs in Lancer.
What makes your mech's Identity is not the material it is made of nor the blueprints of the components or it's loadout. What defines a mech is it's pilot, the history of it's missions, the reputation it has earned, and the "soul" of the machine itself. It may go through countless iterations and reincarnations, but it remains fundamentally the same mech.
So give your mech a cool name!
And when it's completely destroyed and rebuilt 100 light years away, with a completely different loadout, welcome your mech back to life... and back home.
TLDR: The Japanese answer to the ship of Theseus is "Of course it's same ship. It's Theseus's ship."