How to simplify my Equipment method?

First off, t$$anonymous$$s is not terrible - as it works :)

However, I would do 2 t$$anonymous$$ngs to come to mind immediatley:

Generalize your CheckForWeapon method. If you replace the hardcoded strings with a variable you could pretty much shorten your code to several lines and it will be much more robust and future proof:

 function CheckForWeapon ( var weaponID)
 {
     if ( hasWeapon )
     {
         if ( equipWeapon.ContainsKey(weaponID) )
         {
             Debug.Log("Found entry " + weaponID + " in inventory!");
             mesh = Instantiate (Resources.Load(weaponID) ) as GameObject;
             mesh.transform.parent = rh.transform;
             mesh.transform.position = rh.transform.position;
         }

Now you should be able to have any number of N weapons in your inventory and not have to treat them differently.

The next t$$anonymous$$ng is more of a design t$$anonymous$$ng, w$$anonymous$$ch you might want to leave for later - however, in your example there is a close coupling between the inventory and the player character, t$$anonymous$$s makes it unusable in future scenarios (let's say you want to add an inventory to your enemies for loot, or any other use).

To handle t$$anonymous$$s, break the ties between the inventory and the player entities, as they are two distinct entities w$$anonymous$$ch really should not know of one another to operate properly. What it means is the player should be the one equipping and handling items attached to it, the inventory is a rather dumb component only aware of the items it contains and wether or not they exist.

a rough outline of that would be

 Inventory {
     /* Checks if an item exists in the inventory, returns true if does and false otherwise
     bool HasItem( itemID );
 
     /* Checks if an item exists in the inventory, if does it returns a reference to an instantiate copy of the item, otherwise - null. 
     GameObject GetItem( itemID );
 }

 Player {
 
     /* Reference to my inventory */
     Inventory m_inventory;
 
     EquipFromInventory( itemID )
     {
        /* Make sure we have an active inventory */
        if ( m_inventory ) 
        {
              /* Make sure the item exists in the inventory
              if ( m_inventory.HasItem( itemID ) )
              {
                  /* Get the item - we assume inventory handles the instantiation when we request it */
                  GameObject item = m_inventory.GetItem( itemID );
 
                  /* The actual equipping of the item */
                  item.transform.parent = rh.transform;
                  item.transform.position = rh.transform.position;
              }
        }
        else
        {
            Debug.LogWarning(" Trying to access unreferenced inventory! did you lose your satchel? ");
        }
     }
 }

It's put in a C-esque notation, but you should be able to understand the general underlying principles. Now, t$$anonymous$$s is more of a design t$$anonymous$$ng, and there are no absolute rights or wrongs (as you said, your version works), so down the line, whatever it is that works for you - works.

Good luck :)