The Klingon Battle Cruiser looks like it was designed to "cling on" to the saucer sections of Federation Starships. Was that the original concept, and is that where the name "Klingons" came from?
To answer your second question, they were definitely not named because of that capability, even if the capability existed.
From the Star Trek franchise, purportedly by Gene Roddenberry, named for Lieutenant Wilbur Clingan of the Los Angeles Police Department.
Wikipedia supports that etymology of naming them after Roddenberry's former co-worker, sourced from "Alexander, David (1995). Star Trek Creator. Penguin Group. ISBN 0451454405."
As far as being designed that way: I don't know the canon very well, but Memory Alpha doesn't mention such a capability for either 22d century or later Birds of Prey - and the design predates the Federation anyway. From Memory Alpha:
The first encounter occurred on the outer perimeter of the Sol system in early 2153. (ENT: "The Expanse") Starfleet, however, had known of the existence of these ships as early as late 2151. (ENT: "Sleeping Dogs")
This means they were already in service in 2151, the year when NX-01 Enterprise was launched.
No, there is no mention of them being designed to cling on to a saucer section. Also, if you compare the size of the ships, the Klingon ship does not fit around the saucer section in that way. (Trust me - I had all the Trek models as a teen in the 1970s and I actually tried things like that with the models!)
Also, if you read Stephen E. Whitfield's The Making of Star Trek, which was written in the 1960s, before the 3rd season was complete, there were really no such constraints on the design of Klingon ships. Whitfield had direct access to Roddenberry and others in the production crew when writing the book (it was before Star Trek Fandom really even started). He has sections on the Klingons and Romulans and the idea of grabbling a starship was never brought up. They just needed a mean looking ship.
(Added from my comment - comment now deleted)
While producing the original Star Trek, they were under intense pressure to get out each week's episode. Usually a script would be written without the writer having any idea what the spaceships in the script looked like. Unless there were special needs (like grappling hooks or a docking port), the design or structure of the ship would not be specified in the script and the design would be left up to the design department. The script would almsot always be completed before anyone saw a design of any new spaceships in it.
While one may argue that even though the intent of "clinging on" to a Federation ship was not in the first episode with a Klingon ship in it (I can't remember if we saw actual Klingon ships in Errand of Mercy or not), this was also before TV shows were even thinking of story arcs. It just was not done in the 1960s, so the idea of setting up capabilities for later episodes that weren't used in the current episode was almost unthinkable.