Is there anything preventing a Barbarian that is currently raging from using a bonus action to expend another rage and restart the 1 minute duration on their rage?
Jeremy Crawford, Lead rules designer said in a tweet:
The Rage feature doesn't prohibit you from starting a rage while you're raging.
I don't see why not...
So the PHB says:
"On your turn, you can enter a rage as a bonus action."
"You can also end your rage on your turn as a bonus action" PHB 48
So from a RAW perspective I'd probably say no because to enter a rage you most likely cannot already be in a rage. That would be similar to trying to enter a house that you are already inside of if interpreted literally. And I think that the fact that you can forcefully end your rage (essentially exiting it) supports this.
From a RAI perspective I think that it is okay to extend your rage duration by expending another rage slot. Consider this:
"A barbarian can draw on this reservoir of fury only a few times without resting, but those few rages are usually sufficient to defeat whatever threats arise." PHB 46
I believe the emphasis for this lies in that rest is required to replenish a barbarians fury. However, until depleted, that reservoir can continue to be tapped. One does not simply stop being furious because a minute has passed. The main thing to take into account with this mechanic is the limited reservoir of fury.
Consider also, the rage timer could reach its end, kicking the barbarian out of rage at the beginning of their turn, only for them to immediately pop another rage. There is technically no significant amount of time between the first and second rage. It would not make sense for the barbarian to experience a single moment of clarity spanning the bridge of one solitary second before becoming consumed by a furious rage once more.
RAW is most likely a no, but from the perspective of RAI I'd give it a yes.
You don't need to
10 rounds (1 minute) after entering your rage it ends at the start of your 11th turn. The first thing you do with your 11th turn is use your bonus action to start the next rage.
Unless a restriction is stated there isn't one. I.e., Warcasters only have 1 target restriction. All the rage feature states is that you can enter a rage as a bonus action on your turn. It doesn't say "while not raging you can use a bonus action to enter a rage". So if a barbarian is in a drawn-out battle he could on round 10 use a bonus action to continue raging (using that round as the start of the 1 minute, not adding it to the current minute).