Here's a better idea than continuing with Mutt: Set it in modern times and have some entirely new character (maybe Mutt's son or nephew or something) discover Indiana Jones' hat, whip, etc. in a chest in an attic or a museum storeroom (where he works). He asks the now grandfatherly Mutt what all this is. Mutt tells him the tale of "the great man" and this inspires the new character to take up the hat and whip and seek adventures himself. This guy could posses an intangible that is very important -- he's us. For example, I think one of the reasons the X-Files was so successful its first few seasons was that Mulder was a sort of stand-in for all of us who knew something was out there. He was speaking for us. In a modern-day Indy tale, this new guy could be a surrogate or reflection of all us fans who'd gladly take up the hat and whip ourselves. There'd be a stronger connection to the character. The big problem with Mutt, in spite of Shia's good performance, is that he's Lucas' surrogate, he's Lucas' reflection, not our own. Sort of how when M-chlorians became The Force. It was taken away from what we felt it meant to us and Lucas made it his own own own. Sure, he has the right to do that, but we're talking about what makes a magical movie, and part of the recipie for magic is to give the audience a character they can ALL connect to.
So, if someone new discovered Indy's legacy he could pick up the hat and whip and even the name. After all, "Indiana" was a nickname chosen by Indy, so someone new could just pick the same name. Jones would logically be his last name if he was related to Mutt, so there ya go!