No, a monk can't use their Martial Arts feature with improvised weapons. Monk weapons are:
short swords and any simple melee weapons that don’t have the two-handed or heavy property.
Unfortunately, simple melee weapons are weapons that are in the "Simple Melee Weapons" category on the weapons table, and improvised weapons aren't present there. (Otherwise everyone who was proficient with simple melee weapons would be proficient with improvised weapons.)
So improvised weapons aren't simple melee weapons, and therefore they aren't monk weapons either and can't be used with Martial Arts.
I have played monks with Dex-to-damage as well as Wis-to-damage (and Wis-to-attack, for that matter). These things help, undoubtedly, but they are not really sufficient to solve the problems.
The big advantage here is that they allow a certain amount of reduced Multiple Ability Dependency. This helps some of the monk’s serious “number” problems – their HP, AC, attack bonus, and damage are all very low. This helps with those somewhat. But even a monk that gets to use Wisdom for everything still has relatively low numbers, particularly in the damage department. Consider: a chain shirt is +4 AC. To match that, a monk needs 18 Wisdom – about as good as you can hope for starting out. It applies to touch AC, which is (very) nice, but the monk is still ultimately behind here: a small amount of gold allows anyone else to do as well, if not better, than the monk in AC. And then the monk has ¾ BAB, and a relatively small d8 hit die. The rogue (another very weak class in Pathfinder) has the same, but at least gets relatively-substantial damage bonuses from Sneak Attack. So while this helps with the numbers problem, it by no means solves the numbers problem.
Finally, it is critical to recognize that numbers are not the monk’s only problem. The monk has weak numbers, it’s definitely true, but the really big problem of the monk is that it has few if any class features that are both unique and potent. It gets a random mish-mash of class features, most of which are either extremely weak, extremely limited in usage, or both.
And this is largely because the monk is not a clearly focused class: it is trying to emulate too many different fictional characters at the same time. The monk class doesn’t know what it is supposed to be, so it ends up doing a lot of mostly-nothing. The key to “fixing” the monk is to figure out what monk means to your player, and developing fixes that allow the monk to do the thing that the player actually wants when they write “monk” on their character sheet. There are a number of different directions to take it, but it’s key to choose one.
So my advice is to discuss with the new player what she thinks of when she thinks “monk.” There are a few existing archetypes for monk that are decent, including the qinggong monk and monk unchained. Alternatively, another class entirely may model what she thinks of when she thinks “monk,” or there may be more significant houserules you can implement to meet what she wants from the class.
Best Answer
Necessary but insufficient
It would help. It would more than help; I have a hard time imagining them being functional without it. The necklace of natural attacks or scorpion kama are generally necessary for Monks. That said, the necklace of natural attacks does exist (as do similar items in Pathfinder), and it’s not nearly enough to make the Monks good.
The Monk’s problems are problems of design: the people who wrote it evidently had no clear idea of what a Monk was or should do. Thus it receives a mish-mash of random abilities that do not synergize (and frequently contradict one another, see Flurry of Blows and Fast Movement).
To “fix” the Monk, one must first come up with a clear vision of what the Monk is supposed to be and do, and then most likely rebuild it from the ground up focused on that vision.
Or simply use one of the many classes that can effectively model one or more possible visions of what a “monk” should do, without any levels in the “Monk” class. The Cleric and Psychic Warrior are both Open Game Content and quite capable of fulfilling most roles you could imagine for the Monk, for instance. I’d argue that it’s entirely reasonable to treat a Barbarian’s Rage as “Zen Focus” and waive the non-lawful requirement (which I’d further argue is dumb to begin with). Such a “Barbarian” multiclassed with Fighter for combat maneuver mastery and perhaps taking Improved Unarmed Strike would make a decent monk. And if you have Tome of Battle, the Swordsage also does an excellent job.