Why was Frodo "wise" and Samwise a "half-wit"?

Question

According to the sources I’ve found (for example this one), “Frodo” comes from the Old English “fród”, meaning wise. (Apparently in “Hobbit Language” Frodo’s name is “Maura”, but I don’t know if this has significance here or not.)

And “Samwise” comes from the Old English word, well… “samwise”… meaning “half-wit” or “unwise”.

Here are examples of each word in Old English:

þú eart mægenes strang ond on móde fród wís wordcwida
“you are strong in power and wise in your heart, judicious word-speaker”
_Beowulf

Wenað samwise þæt hi on ðis lænan mægen life findan soða gesælða.
“The unwise think they can find true joy in this transitory life.”
_{The Meters of Boethius}

Clearly, their names are exact opposites, but I don’t remember Frodo being particularly wise nor Sam being particularly unwise (but I only saw the movies :P).

Why did Tolkien choose these names?

Answer 1

_{I have gathered some information but this answer will change in time.}

Tolkien did indeed intend the name Samwise to be "half-wit"

Various times throughout his letters Tolkien discusses the name of Samwise, however, while he did intend to name him the "half-wit" Tolkien had a lot of love for the character.

Sam by the way is an abbreviation not of Samuel but of Samwise (The Old E. for Half-wit), as is his father’s name the Gaffer (Ham) for O.E. Hamfast or Stayathome. Hobbits of that class have very Saxon names as a rule....
_{Letters of J.R.R. Tolkien - Letter #72}

this jewel among the hobbits
_{Letters of J.R.R. Tolkien - Letter #76}

Sam is meant to be lovable and laughable.......He is a more representative Hobbit than any others that we see much of: and he has consequently a stronger ingredient of that quality which even some hobbits found at times hard to bear: a vulgarity- by which I do not mean ’down-to-earthiness’- a mental myopia which is proud of itself, a smugness(in varying degrees) and cocksuredness, and a readiness to measure and sum up all things from a limited experience, largely enshrined in sententious traditional ’wisdom’.............Sam is cocksure, and deep down a little conceited; but his conceit is transformed by his devotion for Frodo.
_{Letters of J.R.R. Tolkien - Letter #246}

Tolkien shows great appreciation of the character and while the name does indeed mean "half-wit" I don't believe he intended us to think he though of Sam as stupid, but instead as someone who came from "limited experience". Sam was a much more sympathetic character in Tolkien's mind as opposed to Frodo, and felt in his mind was the far more genuine hobbit:

Sam is the most closely drawn character, the successor to Bilbo of the first book, the genuine hobbit. Frodo will naturally become too ennobled and rarerified by the achievement of the great Quest, and will pass West with all the great figures; but S. will settle down to the Shire and gardens and inns.
_{Letters of J.R.R. Tolkien - Letter #93}

_{Warning the Frodo section is less complete}

Tolkien was again aware the Frodo was from the germanic fród

While intended the Frodo be from the germanic roots to mean "wise by experience" it had further connotations

Frodo is a real name from the Germanic tradition. Its Old English form was Fróda. Its obvious connexion is with the old word fród meaning etymologically 'wise by experience', but it had mythological connexions with legends of the Golden Age in the North.
_{Letters of J.R.R. Tolkien - Letter #168}

Frodo's experienced is shown by his travelling nature in his youth. He had adventured with his two "wealthier" friends, Peregrin and Meriadoc, in his youth.

Frodo went tramping all over the Shire with them; but more often he wandered by himself, and to the amazement of sensible folk he was sometimes seen far from home walking in the hills and woods under the starlight. Merry and Pippin suspected that he visited the Elves at times, as Bilbo had done.
_{The Fellowship of the Ring - Book 1, Chapter 2: The Shadow of the Past}

Answer 2

In Appendix F, section II in The Return of the King, we note the following:

But Sam and his father Ham were really called Ban and Ran. These were shortenings of Banazîr and Ranugad, originally nicknames, meaning 'half-wise, simple' and 'stay-at-home', but being words that had fallen out of colloquial use they remained as traditional names in certain families. I have therefore tried to preserve these features by using Samwise and Hamfast, modernizations of ancient English samwís and hámfoest which corresponded closely in meaning.

The appendix doesn't say anything about Frodo's name specifically, but Edlothiad's answer seems to have enough information about it.

I would have this as a comment, but I think the quote needs better highlighting than merely a comment.