Just wondering what weapon type would be best to give a mage.

The setting here is what I'm calling "modern fantasy"; basically, technology is at modern levels, but there's magic. Furthermore, I'm going to say that this setting doesn't have what I could call "legacy medieval" weapons. In other words, the only weapons the player(s) would expect to find are those that either:
* Are used in modern warfare (say, World War 1 onwarod)
* Are commonly used in violent crime

So, no swords, axes, staves (normally the go-to weapon for mages in medieval fantasy), bows, etc.; but instead we have guns, flamethrowers, grenades, nuclear bombs, and other weapons of that sort. (Of course, knives, bare hands, and things like pepper spray exist and can be used as weapons in this setting.)

Futuristic weapons might be OK in this setting, as long as they're similar to modern weapons and not trying to imitate medieval weaponry (so no lightsabers or similar weapons).

(Magic, of course, exists; otherwise this topic wouldn't make sense.)

So, in this setting, what weapon would you give a mage?

Light submachine gun (maybe Uzi or something similarly compact and portable).

Question: why are you removing staves and axes from weapon lists since in modern setting they could be used weapons for urban combat? Non combat axes are common tools, but are much heavier than proper battle axes; (non magical) staves are used in martial arts. Both of these categories work with violent crime categories.

Rocketlaucher or alternatively artillery gun because mages stay in the back anyway.

Yep , but most mage would pick orbital ion cannon or drone strikes.

How about a baseball bat used as a substitute for a mage's staff? It could even add some humor with the right spin.

Considering that a mage is probably frail and not a trained fighter, they'll probably stay in the back, in cover, if possible. They'll attack medium/long range with spells, other weapons are "if all else fails" and should be light and easy to use.
So maybe a knife as really last resort, and a pistol or maybe light submachine gun (like morolf) suggested for some spray-and-pray.

A shotgun for enemies that come too close might be good too, but there's the recoil to consider. Grenades and stuff take some skill a mage probably won't have.

And could be used as a ranged weapon if it had a boxing glove that sprung out at high speed from one end when a button (located on the handle) was pressed...

A boxing glove with a horse shoe in it? ;-)

Classic: normal guns with magic infused bullets!
Same for explosives or the evergreen magic vials.

No way modern mage is frail , more like a fat blob in one of those american shopping chairs.
and eating cheetos or donuts to replenish mana

9mm pistol or a wand. Knife is acceptable, but you don't bring a knife to a gun fight. A stun gun would work.

Lol, yeah, that should add extra + damage. Oh, and don't forget the obligatory suspended safe, to drop on any unsuspecting enemies straying into its drop-zone...

A gnomehide whip.

Gun. Jack from Mary Skelter takes a supporting role in battles by either healing by shooting them or reducing the insanity stats of your warriors through some sort of tranquilizer injection with the gun. Though I couldnt get into it because of jumpscares.

Books are still valid. If you want, maybe an e-book reader would be a modern equivalent if that character's techy to access spells or tomes. Heck, even a magic UAV or drone could also work like Angelo from FF8.

A pharmacist / chemist / chemical engineer / drug dealer also works as a modern day mage. I remember watching Sense8 and the Indian female pharmacist used her chemical knowledge to make a makeshift bomb. I think there also similar cases in Breaking Bad.