You cannot take an action such as the Cast A Spell action or the Attack action (and thus cannot grapple/shove) but besides that the choice is up to you
Grapple/Shove and casting a spell require specific actions, (the Attack action and the Cast A Spell action respectively) and so we cannot use them when being compelled by crown of madness
The spell simply says we must make a melee attack, it does not specify any further what kind of attack this needs to be but we can look to similar features and their related questions as guidance.
Opportunity attacks, the Hunter Ranger's Whirlwind Attack feature, and the crown of madness spell all use the same wording so we can use previous rulings on the related features to determine how crown of madness should work:
To make the opportunity attack, you use your reaction to make one melee attack against the provoking creature...
You can use your action to make a melee attack against any number of creatures within 5 feet of you...
The charmed target must use its action before moving on each of its turns to make a melee attack...
The answers to the related questions "Can you use a melee spell attack with the Hunter Ranger's Whirlwind Attack?", "Can you grapple/shove when affected by the Crown of Madness spell?", and "Can I use a spell to make an Opportunity Attack?" explain that melee spell attacks and the special melee attacks of grapple/shove are not options for us. Because there are no further restrictions on what kinds of attack we can make all other melee attacks are available to use, in particular these are unarmed strikes, and melee ranged attacks that use weapons (even improvised ones).
The reason we cannot grapple/shove is because these require us to take the Attack action as the Sage Advice Compendium shows:
Grappling and shoving are special melee attacks that require the Attack action (PH, 195).
This is also shown in the linked question above about crown of madness where it is explained that you cannot grapple/shove when under the effects of this spell. This is further supported by the sections on grapple/shove:
When you want to grab a creature or wrestle with it, you can use the Attack action to make a special melee attack, a grapple...
Using the Attack action, you can make a special melee attack to shove a creature...
The reason we cannot make a melee spell attack is because those require the Cast A Spell action as the Sage Advice Compendium explains:
Q. Can you use a melee spell attack to make an opportunity attack?
A. You can’t if the spell attack is created by casting a spell. When a creature triggers an opportunity attack from you, you can use your reaction to make a melee attack against it. The opportunity attack doesn’t suddenly give you the ability to cast a spell, such as shocking grasp.
Each spell has a casting time. A game feature, such as an opportunity attack, doesn’t let you bypass that casting time, unless the feature says otherwise.
This is supported by the other two linked questions above "Can you use a melee spell attack with the Hunter Ranger's Whirlwind Attack?" and "Can I use a spell to make an Opportunity Attack?".
Opportunity attacks and Whirlwind Attack also do not allow us to make melee spell attacks, and as they have the same wording as crown of madness, you cannot make a melee spell attack while under its effects either.
None of these feature specify that we can bypass the casting time requirements of a melee spell attack and so we cannot do this.
Thus we have removed both grapple/shove and melee spell attacks from out list of available options. The only other melee attacks are melee weapon attacks (unarmed strikes, and making a melee attack with a weapon, even an improvised one).
The spell never limits any further what kind of attack you have to make, it doesn't say it has to use a weapon, or that it cannot be improvised and so you can choose to make any kind of melee weapon attack.
That said, this does make the spell weaker; It already requires your action each turn to continue, and the creature makes another save at the end of each of their turns, so it is not likely to last very long.
After those setbacks, allowing creatures to deal only 1 + Strength modifier damage, which is quite pitiful on many enemies, weakens the spell. Its only use would be to prevent a target from using its action in some other way, and though this is quite strong it makes the controlling portion of the spell have nearly no benefit.
Because of this it would be well within a GM's purview to require that the attack actually uses a weapon in its intended way thereby dealing a sizable amount of damage.
A note: In the question "Can you use a melee spell attack with the Hunter Ranger's Whirlwind Attack?" there is a great amount of discussion in the comments and answers about what you can actually do when something allows you to "make a melee attack", there doesn't seem to be any sort of consensus here besides the top answer's flat "no" and so I've used this as a correct answer.