I was looking at the Collins Cobuild Advanced Learner's English Dictionary and I am sure I have noticed many notations?? (i am not sure what they are called) what/that seem to be called PHRASES and along with the definitions, it seems to have some phrases that look to be used with the words I was looking at like the one below. My question is "Can I add some words to the some part of the phrase?". What are the rules or guidelines in regard to one's such an effort? Is this one's such effort or one's such an effort?

From Collins Cobuild Advanced Learner's English Dictionary:

You say on the credit side in order to introduce one or more good things about a situation or person, usually when you have already mentioned the bad things about them.

Can I make it like these?

on the delicate credit side

on the bright credit side

Let's pretend those phrases make sense.
I've always heard (and used) "on the plus side" and not "on the credit side." You can say "to his credit, he is...[and then say something nice about him]."

Perhaps another of the BrE/AmE things.

However, either way, don't add other ajectives to it. Let it stand alone.
No, we don't say on the credit side either.
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Search Google with:
site:literaturepost.com "on the credit side"
and you will find examples from these classics:

Confidence by James, Henry - Chapter 12
Bernard had luminous glimpses of another situation, in which Angela Vivian's
coquetry should meet with a different appreciation; but just now it
was not an item to be entered on the credit side of Wright's account.

White Jacket by Melville, Herman - Chapter 93
... though, truth to tell, some reckless, improvident
seamen, who had lived too fast during the cruise, had little or
nothing now standing on the credit side of their Purser's accounts?

Perhaps some will say that one the above quotations talks about real accounts, and not characters, but I think the expression's applicability could be conceivably extended ...
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Yes, the examples you found are clearly about actual monetary accounts. Or at least tangible assets of some sort.

It is not an expression in use to describe character traits.
There are more examples with "on the credit side."
Here is a link: http://sara.natcorp.ox.ac.uk/cgi-bin/saraWeb?qy=on+the+credit+side