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.
The generally accepted reading of the rules (supported by WotC) is that things do only what they say they do.
Make 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.