Last weekend I played a Khazad Dum army with Balin son of Fundin as leader equipped with Durin's axe. This special weapon gives him a reroll for one die in melee.

There was also a dwarven banner bearer near him, closer than 3". Does this banner grant him an additional reroll, so that in total Balin has two possible rerolls?

It's clear for me, that two banner will not sum up their rerolls, but Durin's axe is not another banner.

Thank you for your opinions or advice!

Yes, You would get an additional re-roll, however you would have to re-roll different dice which could lead you to getting a lower score than you originally intended. For example; you roll a 2 and 5, aiming for the 6 you first reroll the 2 and get a 4, you can then choose to re roll the 5 but not the 4 since that would be a re roll of a re roll which is illegal which would mean you would have to spend more might to get to a 6 should the reroll of the 5 be a lower result.

Hi infinateremains,
thank you very for much for this precise and helpful information!

This is actually how I used him in my last game - not by intent (I actually rarely take banners), but because I took a King's Champion and ended up with a pair of banners to boost things.

As said, Durin's axe grants a re-roll to the duel roll on one die. It doesn't say it acts as a banner for Balin (or Durin when he's using it!) and so there is no reason that a banner can't affect his rolls either - except! As always, you can never re-roll a re-roll. So the axe can re-roll one die and then the banner can only re-roll the other one (Balin in our example only has two attacks, Durin has three though - which means either of his other attacks could benefit from the banner).