Let's consider the role of drones.

In EVE the primary role of combat drones is to allow larger ships to engage ships that are too small for their turrets to track. This role doesn't really exist for infantry, they're the smallest thing out there. It might be useful for vehicles, however, and vehicles having drone bays from which a small number of combat drones can be deployed to harass AV troops would be extremely helpful.

Some ships in EVE actually rely on drones as their primary source of DPS, mostly Gallente (although some of the best drone ships are actually Amarrian), and this could make for an interesting capability both for infantry and vehicles. Gallentean troops may rely heavily on drones to engage enemy troops. This could be in the form of mobile combat drones or sentries, which must are unable to move on their own and must be picked up to be moved.

Ships that make little reliance on drones and are confident in their ability to deal with smaller ships may use eWar drones. These can do basically any type of electronic warfare available to ships. As far as I know the only eWar capability not available to drones is warp drive disruption. How useful would it be to be able to deploy a stasis webifier drone on that escaping HAV? How much would a dropship pilot love to have a flight of ECM drones to disrupt missile locks?

OP mentioned logistics drones. One of the interesting things about EVE's logi drones is that you can't use them to repair yourself. I don't know if that's intended, or simply a limitation of EVE's drone UI, but that's where we are. It could certainly be very helpful to assist your remote repair, or allow you to do emergency spot repairs on an ally you can't get to.

To make it possible to deal with an opponent's drones, they would have to be at least fairly large (about half a meter in diameter, maybe?) and move no faster than an assault suit at a sprint. I would keep their damage fairly toned down, too.

---

But as awesome an addition as they seem on paper, how usable would they be? We have to consider the limitations of our controllers (Damn you, PS3 exclusivity!) and what they actually allow for.

For infantry the obvious thing would seem to be an equipment slot item which has a certain amount of space for drones, and a certain limitation on the amount of drones they can deploy concurrently. Switching to this equipment item allows you to lock on to a target. Continue holding to launch a drone. If balance allows for multiple drones to be deployed at once, continue holding to deploy more. You can then switch back to your firearm and return to the fight normally.

An interesting alternative would be to use a weapons slot, rather than an equipment slot, at least for combat drones. This would also allow for different drone sizes and powers, with small drones, probably eWar types, going in the sidearm slot all the way up to heavy (AV?) drones in heavy weapon slots.

Now with vehicles my gut tells me that drones shouldn't be active modules like they are with infantry, but rather an integral part of the vehicle like they are in EVE. This complicates things, because it means they can't work the same way they do for infantry. We could, perhaps, use the right arrow button, similar to the OMS menu, allowing selection of what grouping of drones to deploy. This leaves the pilot with little to no way to control whom the drones target. Having a special reserved spot on the active mod radial menu is an alternative, but a bit clunky.

One interesting idea (though I do not necessarily advocate it) would be to make drones replace small turret slots, so that the pilot can have passengers control drones by remote.

While I really like the idea of drones I'm not sure it would be good addition to existing dropsuits, due to them then having to be bad enough to make NOT using them a viable option.
On the other hand I'd really love to see them implemented through a new type of dropsuit (maybe even the commander one) or some kind of vehicle (i.e. a drone hive) with which you could actively control more than one for the price of not being able to do anything else while controlling them.

When I think of drones I imagine a hilarious scenario where a full squad assigns their drones to assist one guy (probably a heavy w/ HMG) and then every time a bullet so much as clips an enemy, they've got 5-10 drones bearing down on them instantly.