Can a magical item, have multiple command words to trigger activation? How does this affect the making of the item?
I can see two interpretations of your question. So, I will answer them both.
Do you mean:
Q: Can an item with multiple functions have different command words for each function?
A: Yes. In fact it is assumed since otherwise, you would not know what would trigger when you said the command word. Though, you may argue that this was what created the wand of wonder (rod in Pathfinder).
Q: Can a single function on a magic item have more than one command word?
There is no rules as written method of doing this. Personally, i don't see the point but you you may have spotted something that I haven't thought of.
Conceptually, I don't see why not but anything you do for this would be a house rule. As far as the cost goes, it probably shouldn't be as expensive as adding in another copy of the same function just to get another command word but it shouldn't be free. Custom is always more expensive.
Remember that Identify will give all command words of the item.
This spell functions as detect magic, except that it gives you a +10 enhancement bonus on Spellcraft checks made to identify the properties and command words of magic items in your possession. This spell does not allow you to identify artifacts.