The bard's Bardic Inspiration feature states:
You can inspire others through stirring words or music. To do so, you use a bonus action on your turn to choose one creature other than yourself within 60 feet of you who can hear you. That creature gains one Bardic Inspiration die, a d6.
Once within the next 10 minutes, the creature can roll the die and add the number rolled to one ability check, attack roll, or saving throw it makes.
This basically plays out to be "You can do it!" or "Break a leg!", or just playing their lute to build suspense, to affect one thing at a time.
However, I was thinking about the possibility of changing this up for a more "constant" variation. A smaller pay off (+2, for example) for the entire period, much like listening to your favorite song when doing something challenging; Eye of the Tiger or Danger Zone during a fight, Stayin' Alive while doing a heal or revive check, or just your favorite song to get you in the zone, and your blood pumping.
Additionally, the original time frame is "in the next 10 minutes", which means any time within the next 60 rounds; so a compromise of a constant +2 over say, 1-3 minutes (i.e. the length of your "inpirational boom-box music") would not only fit combat more effectively, but also be more evenly balanced in bonus:duration ratio.
So; would a constant bonus of +2 over a shorter duration (1 minute) be balanced?
Best Answer
Unbalanced and Unconventional
Assuming the Bard still has the same number of Bardic Inspirations per rest this is a much stronger homebrew, and balance isn't the only issue.
Balance Math
For this example, let's take a level 3 Bard with a Charisma of 16. By RAW they have three d6 inspiration dice per long rest. At an average of 3.5 per die, that is a total of 10.5 increase per long rest.
Your variant changes the d6 to a flat +2 benefit, this means for any given roll it will, on average, be less of a benefit than the RAW method. However, by granting a +2 to all rolls for a duration of 1 minute per usage, the total benefit is much higher.
Assuming an average combat length of 3 rounds, any target you use this on is likely to make 3 attack rolls and 1 saving throw per combat (this is conservative). +2 per roll, 4 rolls per combat, 3 combats per day, adds up to a total benefit of +24 per long rest. That is over double the benefit of the RAW method.
The gap only gets wider as you level, at higher levels there are more rolls per round and therefore the constant benefit is even greater.
'But wait!' you say 'this variation doesn't let you choose the important rolls'. Correct, it doesn't, it applies to all of them so you never have to choose if it is more important to use it on this attack, or keep it for your next save. This is power-positive and is unbalanced.
Static Bonuses
It is very rare for a feature to grant a static bonus in 5th edition. A bonus equal to an ability score (e.g. Paladin's Auras), advantage, or an additional dice (e.g. RAW Bardic Inspiration) are far more common. This is part of the design of 5th edition and I usually advise homebrewers to stick within existing examples unless they really know what they are doing.
Complicates Other Features
It is unclear how this would interact with the other uses for Bardic Inspiration, Features like cutting words use Bardic Inspiration dice in a particular way. Would this homebrew rule also apply to the other uses?
It's been this way before
The system you are describing is very close to what exists in Pathfinder 1e and 3.5e D&D. The Bard would spend an action to begin Bardic Inspiration and then could maintain it for free, up to a certain number of rounds per day. While active it provided a static 'morale bonus' to allies within 30ft. This bonus increased as you leveled. There were other were other ways it could be used but they are less relevant.
If you are set on changing this, maybe look into how it worked before and consider how that could be adapted to suit the 5th edition design philosophy.