I'd be happy if you could stay until tomorrow.

Literal

Tomorrow [until-まで] exist-please [if-と] happy [but-のですが].

いてくださると嬉しいのですが — a politely hedged wish. ~と嬉しい ('if you do X, I'd be happy') is wrapped in のですが, which adds the explanatory の and the softening が ('but...'), creating an extremely polite, indirect request.