Right now if your private server expires, you’re required to re-activate it before you’re able to configure it again. That may require paying a decent handful of R$, depending on the game. If the game no longer supports private servers, you don’t even have the option to pay to configure it, it won’t even show up in the game’s server list.
However, even if a server may be inactive, it will still show up in the inventory of anyone who was given permission to join it, under Inventory > Private Servers > Other Private Servers. The biggest example of this would be the Friends Allowed setting, which continues to add the server to the your friends’ inventories. If you add a new friend years after a server has expired, it will still show up in their inventory.
Depending on what the user typed there, they may wish to remove the name of that server. Someone may have named their old server something embarrassing, inappropriate, or even included personal info. Right now though, renaming a private server can be a difficult, sometimes implausible, task.
There are better long-term solutions to this, such as a page dedicated to viewing private servers you have access to, configuring any servers you own, seeing when a server might auto-renew soon, etc. but I feel in terms of quickly fixing up this issue, being able to configure the server without paying R$ to re-open it would be a good solution for now.