Sadly, the user's TZ isn't available to JavaScript. This is why so many sites have you input that as part of your preferences.
You can get the local time with JS, and you can get UTC, and from those two you can narrow it down (you know the current UTC offset, but that's it), but you're not going to get a single answer.
But you're just converting one time to another. If you know their reported UTC time and their reported offset to local, isn't that all you need? For greater precision, ask for their geolocation or use an IP address to narrow down the possibilities, though offset obvious helps.
Now that I think about it, you could auto-suggest based on offset and embed a list of locations per option if they needed to pick between daylight savings or not, for instance. There's room for an interesting widget or service, but it would need to be updated almost as much as a US sales tax calculation widget ;-)
