Hmm. "Channeller"? (As in they channel magic through their weapon. Maybe there is a subclass that does it though their shield instead?)
The literary concept/place in the world will probably be a better place to start with creating a name for the class rather than the mechanics ("I want a gish half-caster")
There are currently 5 or 6 gishy character concepts in 5e at the moment. If you included UA and reskinning divine magic, you might be able to hit 10. Your name and concept needs to distinguish this class from those in terms of their place in the world.
Most of the concepts people are talking about are very much frontliners who enhance their fighting with magic, like the old Eldritch Knight prestige class, the hexblade, the swordmage, and the magus. A spell-based skirmisher should be possible, but I'd call that a variant or secondary build. Definitely more like a fighter than a rogue or monk.
Onto your other questions that you might want to answer when designing this class' abilities, and how I would answer them:
What would they be doing round-by-round in combat? Hitting people with magic-enhanced weapon attacks.
Are they buffing themselves then making weapon attacks? yes; it's what they do - Extra attack and bonus to concentration saves perhaps? Both are good ideas
Do they stand toe-to-toe with enemies in the front line like a Fighter? Yes Or more mobile and hit-and-run like a Monk? Not normally, but making this an option is better than not.
Are they creating a spell-like effect (even if it is produced from their weapon) most rounds? yes - More spell slots or weapon-based cantrips. a mix, for balance sake.
Are they combining weapon attacks with spells? (Such as lightning lure or Misty Step then making weapon attacks) - Something like the Eldritch knight's ability to make bonus action attack after casting a spell/cantrip. No, and I think this is important: the concept isn't you cast *then* attack - casting and attacking are the same thing. You can already make a character who alternates.
Or do their weapon attacks have the effect of spells? (Smite spells, creating Thunderwave by striking the ground, throwing your weapon along a line hitting everything in it's path, blasting fire out of your weapon as you sweep it in an arc etc.) Generally yes, although I'd probably phrase it as "their spells include weapon attacks" since it would mesh with the rules better. - Spell selection and reskinning. The spells needed to realize this don't really exist yet - some do, but a couple dozen new spells would be needed.
What would the class be doing out of combat? Depends on the subclass, but generally caster stuff.
Do they even get spells that aren't focused out of their weapon? Can you see this class casting comprehend languages? Charm person? How about Fly or enhance ability? Absolutely - they're still magic-users, and no magic-user is restrited to combat-only spells.
Do they have Ritual casting? Arcane ones do, but other magic traditions might not.
How many skill proficiencies do they get, and what skills? Normal two, but with a fairly broad selection: athletics, any of the int skills, probably intimidation.
Is this class particularly good at something out of combat? Evaluating opponents? Spotting magical traps? Detect magic is definitely an option, and other utility spells would be available.
If you have four players sitting down at a session zero to come up with a balanced party, and one says that they'd like to play this class, would the other players take it as the "Offensive caster" slot being covered, or the "melee front-liner" slot? Or something else? You're a frontliner, generally - like a fighter or paladin, but with a different array of secondary options. Of course you could focus yourself in a different direction, but that's a specific choice.
