The first question is whether or not the next Indiana Jones will be an actual reboot or merely a continuation of the original series. Indiana Jones and the Crystal Skull had our hero going off with the woman he always loved, leaving his beloved fedora to be donned by the next generation. One possible route for a new Indy would be a simple carrying on of the name for another adventurer/archaeologist. This would necessitate a certain amount of retconning the final installment, as we can be reasonably certain that Shia LaBeouf’s character Mutt will not be the new Indy. But advancing the narrative into the 1950s and possibly 60s, rather than rewinding and writing over the original, means that the next films can maintain a nice singular narrative as well as bringing Indy into new eras. It would be respectful to the original series and keep open the possibility of Harrison Ford’s return as the original Dr. Jones, perhaps irate that some new guy is trading on his name.
Beyond simple retconning, the problem with carrying on with the series would be explaining why the “new” Indiana needs to take on the name of the old. This has been an argument about James Bond before – but Bond’s name is easily written off as a code name, and thus the series has some basic (though not exclusive) continuity with shifting actors and time periods.
Indiana Jones is not a secret agent, or any agent really – he’s first an archaeologist and an adventurer, often not working for anyone save himself, his museum, and in the interests of history. Unless we put it down to some new young man taking on the name as a way of advancing his own career – which in itself would need some clever writing – trying to create another Indiana Jones in the first’s timeline would difficult, if not impossible.
A proper reboot of the entire franchise is another option. This is perhaps more flexible than a mere sequel, allowing Spielberg and his new cast to search for different adventures for Indiana without having to worry about being true to the old ones. The new series could pick up back in the 1930s and 40s, or move forward to adventures in the 50s and 60s, without having to deal with the old times. It also allows for a brand new interpretation of the character.
Really, the sole problem with this direction would be a sense of actively ret-conning the entire franchise. Unlike the Bond franchise, which had original novels to turn to for many of the early films (and some more recent ones), the Indiana Jones franchise is solely a cinematic endeavor. Do we really want to see Spielberg and Lucas’s original vision written over, even by the men themselves?