Its not a general solution, but a USB to RS-232 interface cable has a high probability of being built out of either a Silicon Labs CP210x chip, or a close relative of the FTDI FT232R chip.
You can get driver packages for both easily on-line, and the odds that one is right are pretty good.
Finding VID, PID, REV, CLASS, SUBCLASS, PROTOCOL etc.
In the general case, you should be able to extract the VID/PID and device class codes from the node in the device manager corresponding to the unknown device. VID and PID can be read from the Details tab, by selecting Hardware Ids, and the Class and Subclass by selecting Compatible Ids.
VID/PID/REV
The VID can usually be translated to an actual vendor name since vendors are supposed to buy VIDs and register with the USB-IF. The list linked by Dennis is kept reasonably up to date, and may be the quickest way to just look up a vendor name.
Windows will match a driver as specifically as it can, so a vendor can bug-fix a product without using up one of their precious PID codes by modifying the revision code, and produce a driver that compensates for the bug in old devices by matching it to the triple of VID/PID/REV.
CLASS/SUB/PROTO
If the device implements one of the standard device classes, then the VID/PID don't matter nearly as much as the class codes. This is why nearly any USB thumb drive "just works"; they all promise to implement the Mass Storage Class, and Windows knows that means they act like a disk and should have a recognized partition table and file system.
There is a list of the class codes about 90% of the way down the list linked above. Don't be surprised to find a lot of devices that say CLASS 0xFF, which is the vendor-defined class.
Part of your problem is that there isn't a completely obvious standard class for an RS232 bridge device, and it doesn't look Windows has nominated a right answer with a stock driver either.
The classes in which you might put a USB to serial translator got standardized relatively late in the game, not all flavors of Windows have a stock driver for it, and not all of the devices claim to implement it.
Best Answer
Ditto to oneguynick's answer. I don't know the HP one, but the EVGA+ adapter I have uses the DisplayLink driver. From what I've seen online, most USB display adapters are powered by DisplayLink.