Looks like rain in Varenna

We had a rainy day in Varenna when we were there.

That was the day we took the ferry to Menaggio, bus to Lugano, train to Como, and the hydrofoil back into Varenna that night.

We carried umbrellas ! Hoping the sun comes out though !

Is a day trip to Milan worth the time on the train?

Sure. It's only an hour away so you could do quite a lot in a day.
Editing to add: Bergamo is another option. The old part of the city - Città Alta - which is perched above the newer part (Città Bassa) is a compact, wonderful maze of Medieval architecture and a bit of rain would just add to the atmosphere. It's also only an hour away by cheap, regionale train (you'll change trains in Lecco). You can reach the old town via city bus #1 from the train station. See these links for info on the town, bus route, tickets, etc. You can buy individual tickets or a 1-day pass at the train station: validate upon boarding the bus:

http://www.visitbergamo.net/en/

https://www.atb.bergamo.it/en/atb-network/routes-and-timetables

As you're probably aware, Varenna's train station doesn't have ticket office so in either case, purchase those at the nearby travel agency and just buy round-trip so you have them in hand to come home. And don't forget to validate them each way before boarding the train!

Next week is a longish time from now in terms of accurate weather report. And even if it does rain, it may not be a total washout.

Predicting weather a week out is more art than science, I think. I remember being in Assisi in late May and Accuweather said it was sunny, when rain was covering the entire peninsula for days.