I'm actually doing it right now for a mod, and it's a lot easier than you'd think. I don't have a copy of the CS on hand, so this is partly from memory.
First of all, you handle dialogue via the quest editor, not the dialogue option in the character editing window. You'd start a new quest-- maybe name it something like "CompanionNameStory."
Here's what the quest editor window looks like: Like this: http://finalgirl.us/Oblivion/valv2_VariTkTest.png.
You add topics in the column to the left marked "Editor ID." I'm not using a particularly good naming convention there, but you get the idea.
"ValLT01" leads to the PC response of (Ask him to continue). The parenthesis mark ( makes that topic appear at the top of the dialogue chain.
Your NPC response goes in the "Info" section, and can be no longer than 150 characters per line (I think) although you can add as many lines as you want in "Response text". When you add a line of dialogue, you also decide what kind of facial expression your NPC will have as s/he speaks.
If you only want the dialogue to be said once, check "say once;" if you want the conversation to not return to the menu, check "goodbye." If you have multiple responses (as I do), you can check "random" for, well, a random response each time you speak to the NPC.
Since you're doing this with Toaster's mod, make sure you use a condition so that you don't end up with every NPC using the dialogue. See the blue highlighted line at the bottom? That guarantees that only the character with that ID will use the dialogue.
And that's basically it. As far as I've figured, anyway. Hope it makes a bit of sense.