I am homebrewing a cursed magic item, and I'm wondering about my options for how the curse can be activated.

• By using the item without attuning first?
• Only by first attuning to the item?
• With a saving throw?
• Without having the player roll a saving throw?

In other words, could a PC pick up an item, see that it can be "activated" in some way, choose to activate it (prior to one hour of attuning) and be cursed?

Or should there always be a save against being cursed? Maybe especially when the curse can happen even if the PC doesn't choose to attune first.

Almost all standard cursed items activate as cursed on attunement and stay that way until a remove curse spell. It’s really up to you how you want to homebrew it though. It’s probably simpler to use those standard rules, but if you have players who are more experienced or do a lot of meta-gaming, you might want to change it up to keep them on their toes. 

Thank you!

So you are saying typically attuning activates the curse, right? In that case, is there a Saving Throw?

If I do an immediate attuning, should that include a Saving Throw?

Generally there is no saving throw to avoid the curse. 
I’m not sure what you mean be immediate attuning, and again, it’s always up to you. That said, I would not allow the save. The curse isn’t revealed until the end of the process; it’s not like there’s a point 55 minutes into the hour where the character realizes what’s going on and can choose to stop. Once they finish attuning, the curse reveals itself (or doesn’t, it can depend) so the amount of time it takes to attune seems irrelevant, to me at least.

The DMG lets you sit with an item over a short rest to identify the properties of the item. My group has decided that if you try that with a cursed item, at the end of the rest you become attuned and cursed. Same result if you try to wear it or use it to figure out the properties.