I have some irreverent suggestions here:
1. Being able to release singles in other countries. You know how in-game there is different audiences to please in order to get good sales? How about if each country had a preference of certain audiences in it? For example if i made a song that has a high percentage towards casual fans, It would be neat to be able to also release the song in the UK because there is a preference towards casual songs there but it would not be a good idea to release it in South Africa because there is a low preference for casual songs there and it might be a good or bad idea to release it in Brazil because they have a neutral preference with or against casual songs. (Basically they're in the purple) Releasing songs in other countries not only could increase sales but gains more fans within that country so when you tour there you could get a boost of some sorts.
2. Since the game is in beta I'm going to assume this is going to be added in future versions but charting would be nice, no matter how high or low my sales are my chart position is always "n/a".
3. I think it would be pretty interesting to have sabotage available. Is there an girl group that outsold yours? No problem! Just frame their top idol, cause a fake scandal, vandalize their practice rooms, steal their songs in the making, or even send a spy to learn their tactics. It would be pretty fun to ruin other rival groups as well as fending off sabotage attempts towards yourself.
4. Likitty1227's idea of an international trait is a pretty fun idea, I'd love to have some diversity in the group but I'd also like to add a little bit of details to it. Like if a player really wants a Taiwanese or Thai member specifically then we should be able to hold international auditions in certain countries to get specific nationalities. Also it would be cool to have a system of some sorts that allows idols to learn other languages that will give them boosts when touring internationally or on internet shows, maybe if there's an international idol who's native language is the language that the domestic idol is trying to learn, she could learn the language faster.
5. I'd like it if there was more detail toward's the song making and formation process. Like we could adjust the line distribution to improve or harm relationships towards idols, it could also boost sales if high ranked idols get more lines than lower ranked idols. Being able to release a music video would be pretty fun. Collaborating with other groups or artists would be fun and could help boost fans.
Edit: 6. I like Yurei Okami's idea of adding mini games to concerts or something related to that. I find concerts to become a little boring after awhile, you click start, choose cards, click next song and maybe you'll get an event. Maybe you could add a fast forward or skip button for when a song or talk break happens?
That's what i got for now but idk I'm just a random person on the internet who spent $25 on this game.