ED-Ecated
Found all of E-DE's upgrades in the Divide
We know that ED-E was supposed to be delivered to an Enclave military base. This probably suggests that a location in LR might be an Enclave base or that we'll meet up with some characters who have a connection to the enclave. Another possibility is that we might see some other creatures that are like ED-E that we can get the upgrades from.
Condemned to repeat it
Decided the fate of all the Divide dwellers
The name suggests that we're going to be dealing with the history of the divide. There's a quote that goes something like "Those who do not understand history are condemned to repeat it." Whatever happened to the divide in the past, the courier might end up doing again. We know from another achievement that there are several warheads in the Divide. Perhaps they have something to do with what happened to it.
Hometown Hero
Complete lonesome road
A lot of other people have been assuming that this provides the character with some back story, more specifically suggesting that the courier originally came from the area. However, it's possible that this could refer to Ulysses, another central character in the DLC.
Rocket's Red Glare
Fully upgraded the Divide's signature weapon
A lot of what we know about Ulysses relates to pre-war America and this achievement title appears to be pull straight from the star-spangled banner, possibly suggesting weapons that are connected in some way with the pre-war US of the Fallout universe. It could also be a reference to a large number of explosive weapons found in the DLC as other posters have suggested.
Warhead Hunter
Detonated all of the warheads in the Divide
This more or less certainly implies that there are nuclear warheads in the Divide and that they play an important role in the story. In OWB we learned that Ulysses was interested in meteorological data at Big MT, possibly suggesting a connection between these warheads and the weather. Given the condemned to repeat it quest, we may have to make a decision between detonating the nukes, essentially repeating history, or making some other decision.