Solution 1:

Going by the book: RFC 2616-9.7:

  The DELETE method requests that the origin server delete the resource 
  identified by the Request-URI. This method MAY be overridden by human 
  intervention (or other means) on the origin server. The client cannot
  be guaranteed that the operation has been carried out, even if the 
  status code returned from the origin server indicates that the action
  has  been completed successfully. However, the server SHOULD NOT 
  indicate success unless, at the time the response is given, if it intends
  to delete the resource or move it to an inaccessible location.

When you DELETE a resource the server should mark the resource for deletion on it's side. It doesn't really have to delete the resource, it just can't give any guarantee that the operation has been carried out. Even so, the server shouldn't say it's been deleted when it hasn't.

  A successful response SHOULD be 200 (OK) if the response includes an entity
  describing the status, 202 (Accepted) if the action has not yet been enacted,
  or 204 (No Content) if the action has been enacted but the response does not
  include an entity.

If the operation is delayed, send a 202 and an entity body describing the result of the action. (Think of a poll-able "task" representing the server's deferred deletion of the resource; It could theoretically leave it forever in that state.) All it has to do is prevent the client from retrieving it again in it's original form. Use a 410 for the response code, and when the "task" finishes or the server otherwise deletes the resource, return a 404.

However, if a DELETE's semantics don't make sense for the resource in question, perhaps it's not a deletion you're looking for, but an addition state transition that alters the resource state but keeps it accessible? In that case, use a PUT/PATCH to update the resource and be done.

Solution 2:

The Short Version

You cannot RESTfully undelete a resource using any method on it's original URI - it's illogical, because any operation attempted on a resource that has been deleted should return either a 404 or a 410. While this is not explicitly stated in the spec, it's strongly implied in the definition of the DELETE method 1 (emphasis added):

In effect, this method is similar to the rm command in UNIX: it expresses a deletion operation on the URI mapping of the origin server rather than an expectation that the previously associated information be deleted.

In other words, when you've DELETEd a resource, the server no longer maps that URI to that data. So you can't PUT or POST to it to make an update like "mark this as undeleted" etc. (Remember that a resource is defined as a mapping between a URI and some underlying data).

Some Solutions

Since it's explicitly stated that the underlying data is not necessarily deleted, it doesn't preclude the server making a new URI mapping as part of the DELETE implementation, thereby effectively making a backup copy that can be restored later.

You could have a "/deleted/" collection that contains all the deleted items - but how would you actually perform the undelete? Perhaps simplest RESTful way is to have the client retrieve the item with GET, and then POST it to the desired URL.

What if you need to be able to restore the deleted item to it's original location? If you're using a media type that supports it, you could include the original URI in the response to a GET from the /deleted/ collection. The client could then use it to POST. Such a response might look like this in JSON:

{
    "original-url":"/some/place/this/was/deleted/from",
    "body":<base64 encoded body>
}

The client could then POST that body to that URI to perform an undelete.

Alternatively, if your resource definition allows the concept of moving (by updating a "location" property or something like that) then you can do a partial update and avoid the round trip of the entire object. Or, do what most people do and implement an RPC-like operation to tell the server to move the resource! UnRESTful, yes but it will probably work fine in most situations.

How You Decide These Things

Regarding the question of how you decide these things: you have to consider what delete means in the context of your application, and why you want it. In a lot of applications, nothing ever gets deleted, and "delete" really just means "exclude this item from all further queries/listings etc. unless I explicitly undelete it". So, it's really just a piece of metadata, or a move operation. In that case, why bother with HTTP DELETE? One reason might be if you want a 2-tiered delete - a soft or temporary version that's undoable, and a hard/permanent version that's, well...not.

Absent any specific application context, I'd be inclined to implement them like this:

I don't want to see this resource any longer, for my convenience: POST a partial update to mark the resource as "temporarily deleted"

I don't want anyone to be able to reach this resource any longer because it's embarrassing/incriminating/costs me money/etc: HTTP DELETE

The next question to consider is: should the permanent delete only unmap the URI permanently, so that no one can link to it any longer, or is it necessary to purge the underlying data too? Obviously if you keep the data, then an administrator could restore even a "permanently" deleted resource (not through any RESTful interface however). The downside of this is that if the owner of the data really wants it purged, then an admin has to do that outside the REST interface.