Specifically, if someone lobs a bead from a Necklace of Fireballs, can Mage Hand block or catch the bead before it reaches the target?
Maybe, if you ready your action to do so
I'm going to assume that you already have mage hand cast so that casting time is not an issue.
By default you have no ability to control mage hand outside of your turn. Since the attack is not happening on your turn you have no opportunity to intervene. The caster cannot react to an object being thrown, but what if there was a way that they could react to it?
The Ready Action allows you to react to a specific "perceivable circumstance". If you had some reason to believe that someone was going to throw something at you, you could take the Ready action on your turn with the trigger: "If they throw something at me, I'm going to use my mage hand to catch it."
This consumes both your action and your reaction but does achieve the goal you were looking for. You still require a lenient DM for this to work however, as counting "catching an object" as "manipulating an object" is somewhat open to interpretation.
What about the fireball?
So we have established that with some DMs you may be able to use a readied action to catch a bead from the Necklace of Fireballs. But what happens to the fireball when we do?
The trigger for the fireball from the item's description is:
When it reaches the end of its trajectory
Unfortunately "trajectory" is not a game term used anywhere else as can be seen from this search. Therefore it is up to the DM what exactly counts as reaching the end of its trajectory. I can see two possible interpretations.
The bead detonates wherever it stops. Trajectory is defined as "the trajectory of a moving object is the path that it follows as it moves." Notably is the "as it moves" part. Therefore the ends of the objects trajectory is the place where it stops. The fireball will be trigger at the place where your mage hand intercepts it.
The bead only detonates at the end of its intended trajectory. When thrown, the bead is given a trajectory to the place where the user intends it to go. Only reaching the end of this trajectory will detonate the bead, therefore intercepting the bead prevents the fireball entirely.
This option then exposes questions about what happens to the bead after it is caught. Does it become inert? Would moving it to the end of its original trajectory trigger it later? In my view these question weaken this option as an appropriate ruling.
I heavily favour the first option and that is how I would rule at the table. I believe it is most in keeping with the intent of the item. Ruling this way makes the point where you intercept the bead critical, additionally it doesn't allow a cantrip to completely negates a rare wondrous item.
Technically gray, but I would allow it – under some conditions – for the "rule of cool"
The mechanics of mage hand are pretty vague in D&D 5e, and the examples given in the cantrip's text don't seem to be exhaustive. It mentions:
You can use the hand to manipulate an object, open an unlocked door or container, stow or retrieve an item from an open container, or pour the contents out of a vial.
Is this all it can do? I don't think so, because it adds later that
The hand can't attack, activate magic items, or carry more than 10 pounds.
The description remains generic on the things you can do, but is very strict on what you cannot. Could the mage hand hold a light living creature, like a mouse? It's certainly not an object, but it wouldn't make sense to forbid it. As I interpret it, it's a phantasmal hand with very little strength (therefore no attacks or any effort beyond 10 pounds) and which can't complete complex tasks (therefore no activating magic objects); beyond that, the player's fantasy's the limit, and it should be rewarded (by the by, it's pretty easy to balance any situation when the cursed orb of death weighs, wouldn't you know it, 11 pounds).
Therefore, catching an object with the mage hand would be pretty practical: it's definitely in the scope of manipulating an object and it's something this telepathic appendix should easily accomplish.
But what if it's somebody else's turn?
This is the real deal breaker. And naturally, the most likely instance. The only way for a player to do anything when it's not their turn is through a reaction, which normally consists solely of the opportunity attack. As mentioned above, you could also use the Ready action to morph your reaction into pretty much whatever you want, but preparing for enemies to throw stuff and catch it in their tracks would be a waste of an action 99.999% of the time.
Since, to the best of my knowledge, the manuals don't specify that the reaction can only be changed into something different than opportunity attacks (not counting class abilities, etc.), I would rule to use it for many improvised uses too small to consume a Ready action, like answering somebody's question (the alternative would be to wait for your turn, when the answer could be irrelevant), catching staff thrown at me (the alternative would be to wait for my turn, when it could be useless), grab on to a ledge if the tower we were fighting on was collapsing (if it wasn't a combat situation, as a DM you would allow your players for a similar roll, right? So why disallow it in combat, where your senses are heightened and, as the PHB mentions in the table concerning hiding, most creatures stay alert for signs of danger all around?).
The reaction is the perfect tool to react to simple, not exactly combat-related, urgent situations, and the game without these sorts of malleable mechanics to be interpreted in a creative and plausible way, would be worse off.
I know I'm pushing the text here, and that all this interpretation could be refuted by a stricter DM (again, see above), but given that I'm not openly contradicting any rules and that this interpretation only makes the game more interactive, why would anyone do any different for stubbornness's sake?
So to answer the original question, if you had your mage hand active and your character couldn't reach it but the hand could, I'd allow the player to use his reaction (if still available) to manipulate the hand and intercept the stuff in midair if it flied in its proximity, probably with an Intelligence (Sleight of Gand) check if the hand wasn't exactly at the end of the trajectory. If any of these conditions weren't fulfilled, I'd require a Ready action, which I know is tantamount to forbid this interaction.
The generally accepted reading of the rules (supported by WotC) is that things do only what they say they do.
Mage hand can “manipulate an object, open an unlocked door or container, stow or retrieve an item from an open container, or pour the contents out of a vial” - and that’s all.