Honestly, I think you don't have to create your own months/days naming convention. Or minutes/seconds/hours. At least... not unless you're doing something sci fi, where the lengths of things matter.
The problem you'll run into is that there's little way to know what is or is not the "beginning of the week", nor will you really care what the month is.
Final Fantasy XIV tried to do this with "picking a birthday". Without living in that world as a real person, the months and days didn't mean anything to me. In the context of a video game, you're going to have the same issue. Does "Sunday" mean anything to a video game player? Not really. Not unless you have some kind of event or something the player cares about every Sunday. So, changing the names of these days isn't going to have much context either.
The last game I played where the Calendar mattered was Stardew Valley. Let me show you how much that actually mattered:
1. On Sunday, I can begin giving gifts to people if I'd given them two gifts last week. I only bother remembering this after I've completed most of the game and have begun giving gifts to get the last of the recipes.
2. Pierre's is closed on Wednesdays, which I have never remembered in the 400 hours I've played the game until I just happened to need seeds on a Wednesday and he was closed so I couldn't get them.
3. On Tuesday, many of the women of the town go to Pierre's to exercise, so their businesses are closed. I forgot this as often as I forgot about Pierre's.
4. On Friday and Sunday, there's a trader in the forest who sells random stuff. I always forget about this unless I happen to walk by them.
5. On Sunday and Wednesday, "The Queen of Sauce" airs for recipes. I remembered this as I didn't want to wait to try to get recipes I've missed. I also did not want to miss any recipes.
See, unless we live in the world we're interacting with... the names for days, weeks, months, seconds, minutes, and hours... don't really mean anything to us.
"What month is it?" "Oh, it's Cynbalarium." "Okay. Is it any different from Querislaus last month?" "In terms of the game? Nope. Just flavor text." "Okay, then I'll continue to not remember what the month is since it doesn't matter."
Meanwhile, in the real world... "It's October" "Oh, Halloween is this month, and the leaves start turning colors and falling from trees... it's starting to get a bit colder... my kids are gonna want costumes... candy will go on sale for cheap..." "Next month is November." "Yep, gonna go visit relatives on my wife's side of the family for Thanksgiving. I hope we don't get early snow. I wonder who's going to win the big game."
Do you see the difference?
Unless you give some pretty good context for a player, they really don't care what you call these things, as these things honestly don't matter anyway. It's like if you renamed grass to "Slish". I mean... nice you came up with a name... but... it's grass. I don't care.