See, for every parameter and identity we provide denote them with some letter. Now in the case of Flux Density it was done by B. There are some popular thoughts about it.
One, to show respect for Mr. Biot for his Biot Savart Law. But some claimed that the word B was used even before Biot Savart Law.
B stands for Flux Density and if we focus on flux density for a moment, the magnetic lines or magnetic lines of forces that travels from north to south as far our statements or assumptions or whatever you may call it.
Even before Mr Biot, during 1600 Mr. William Gilbert in his "De Magnete" refers the poperty of magnet to have Boreal and Aureal (North and South) property.
So that was the reason to take B from Boreal
Boreal means "relating to north" or "coming from north" so as the magnetic lines comes from north upto the south pole.
And arguments for describing H for Magnetic Force. I myself have no clue why it is so.
But as everywhere we have relations between "energy" and "Field Intensity" so Magnetism demands the same kind of relation. We already knew the relation between B and H.
F is already used as force elsewhere, J is already Current Density so that might be the reason to rely on H I hope.