First off, I’m assuming you have read the article “Multithreading and the Entity Framework” on MSDN.
Solution #1 is almost certainly the safest from a threading perspective, since you are guaranteeing that only one thread is interacting with the context at any give time. There is nothing inherently wrong with keeping the context around- it isn’t keeping database connections open behind the scenes, so it’s just memory overhead. This could, of course, present a performance problem if you end up bottlenecked on that thread and the whole application is written with single-db-thread assumptions.
Solution #2 seems unworkable to me- you’d end up with subtle bugs throughout the application where people forget to re-attach (or detach) entities.
One solution is to not use your entity objects in the UI layer of the application. I’d recommend this anyway- chances are, the structure/layout of the entity objects is not optimal for how you want to display things on your user interface (this is the reason for the family of MVC patterns). Your DAL should have methods which are business logic specific (UpdateCustomer for instance), and it should decide internally whether to create a new Context or use a stored one. You can start with the single stored context approach, and then if you run into bottleneck issues you have a limited surface area where you need to make changes.
The downside is that you need to write a lot more code- you’d have your EF entities, but you’d also have business entities that have duplicate properties and potentially differing cardinality of many of the EF entities. To mitigate this, you can use frameworks like AutoMapper to simplify copying properties from the EF entities to the business entities and back again.