I think the easiest way to do this would be to allow her some sort of domination and control over a Swarm of Insects from the MM. Allowing her to shape it for flavor but I would keep the same damage. Reskinning the given monster to suit her needs having it assume cartoonish shapes so she can make her puns in game. Sounds great to be honest. 4d4 Piercing is the damage from the book.
Now the question is does she want to summon these magically or cultivate them with Bee Husbandry like the mad druid from Keep on the Borderlands. In the case of the latter it could be her downtime activity, but they would obviously be a limited resource, albeit an extremely intimidating one. Sneaking up on your enemies would be hard due to noise, or maybe she has them hug her under her cloak and explode out? That would sure freak me out. Or even use a version of Dominate Monster to control the swarms even temporarily. Maybe even have her have researched a spell or one granted that allows her to control bees like level 2 for Dominate Insect Swarm.
"Beeing" an Eldritch Knight sort of causes issues with my above ideas, since they are initially and mostly limited to Abjuration and Evocation, but maybe when she is able have a low level variation to work toward for Insect Plague that allows for it to only be bees and mobile but limit the damage.
Another really old idea posted in a couple of Dragons for previous editions (which works for any edition) would be just give her evocations a flavor of beeing bees. Like her Fireball could do Piercing damage and explode into a short lived swarm, that could be her signature schtick. This could be a feat probably to change damage type of spells a certain number of times per dawn or so.