Items gain prestige from two ways, I believe. This comes from about 3 years of observation. I'm fairly confident I'm correct here, but as always, I'm interested to know if there are any lesser known features.
1. A noble holds it and goes through battles with it. The conditions that lead to prestige increase from battles are unknown to me. Not all battles lead to prestige gains, and there doesn't seem to be a clear connection between quantity of battles and prestige gains. Possibly size of battles? There exists some vague "glory" value which might play a role possibly.
2. An adventurer holds it and meets a sage who can improve the item. This is relatively new and probably uncommon. This is the only way I know about that an adventurer can directly increase the prestige of an item, and even have it imbued with special abilities.
Now for naming. The convention looks like the following: <Prefix> <Item Name> <Suffix>
Note that it is not necessarily the case that the prefix and suffix have to contain only one value, or any values at all. Of course, it is exceedingly rare to see something called simply "Sword", and it might in fact be impossible if there is a condition set for all items that it needs at least one value in either prefix or suffix. Item name also does not refer to item type, but the actual name of the item, like "Sword", "Book", "Scroll", "Saber", etc. There appear to be a lot of prefixes and suffixes as well, and with more than one prefix/suffix value for any given item, there are quite a few possibilities, such that even if a given name is taken out of the random generating machine after it's discovered, it is probably still unlikely that the same item name would pop up without any restrictions on naming.