I would let players lose a skill for weapon proficiencies.
This was my first thought as well.
A separate full "point buy" system might be an interesting idea, it will lead to another optimization mini-game like the point-buy for ability scores or the skill points in 3e. Whether this is good or bad depends on your players' interests and whether they like fiddle bits: that's because weapons are not all equivalent, so at the very least you'd have to differentiate the value between simple and martial weapons (and if you investigate the weapons list further, it might still not satisfy you and you'll want to also give different points value to different weapons in the same group).
I am not even considering armors yet, because those are even more complex considering there is a hierarchy to satisfy: light before medium before heavy. So what are you going to do here? Would you allow to buy proficiency in a single heavy armor without having lesser armors proficiency? Would you require at least one, half or all the proficiencies in the lower tier? There are many possible rules here, each with pros and cons.
Still, in a hypothetical new game design it would be generally a viable system. On the other hand, I wouldn't like to apply such system on top of 5e, mainly because for 5e martial classes there is nothing to gain and everything to lose from a point-buy system, since those classes already have all weapons proficiencies.
Spellcasters can completely avoid weapons if they want, since they can choose attack cantrips. Low-HP/AC casters who prefer to use a weapon typically IMXP choose the light crossbow (or longbow if elves) because it's ranged. High-HP/AC casters are more often melee-oriented, as they also partially contribute to front lines. The majority of casters however will be more interested in optimizing their spells rather than weaponry.
Generally I kind of think that a point-buy system would be an overkill compared to its usefulness. A very optimization-oriented player would like the option between many lesser proficiencies and less better proficiencies, but most players would just choose ONE favourite weapon and stick with it forever. The drawback is having to give up magic items found in treasure that you can't use, but this depends a lot on what are the DM's decisions on magic items.
So besides the amusing theorycrafting, I don't see much practical benefit in a whole system for this... if an occasional player wants a different weapon, I'd rather look into swapping an existing proficiency for that. Indeed the PHB offers only a feat option for acquiring more weapon profs, which is quite expensive. Swapping a weapon prof for another is too generous (everyone would swap a simple weapon for a martial weapon), unless you restrict it to the same weapon group but then it will be useless. Giving up a skill for a martial weapon proficiency sounds still expensive but almost right to me.
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