If you use 'more,' 'variety' is noncount:
We need more variety on the menu / in the selections.
(= mehr Abwechslung?)
If you use 'bigger' (informal) or 'greater' (better for writing), 'variety' is usually count:
We need a greater(/bigger) variety of dishes on the menu.
(= eine größere Vielfalt/Auswahl?)
There's also another meaning of 'variety' used in agriculture, gardening, etc. Here 'bigger' is informal and 'larger' is standard.
This is a larger(/bigger) variety of pumpkin. It can grow up to 1000 lbs.
(= eine größere Sorte/Varietät?)
If that doesn't answer your question, please give some context.