If the account in question has been sitting dormant for a number of years (date last active and date last reported) then I would leave it alone. Paying off the debt could cause those dates to be updated to this month, thus making it fresher and potentially causing a ding to your credit score.
I can agree with this especially since you're applying for a mortgage. If you pay it, it will show as a paid debt but it would recycle the DLA.
Generally speaking, most lenders won't bat an eye at a single collection of $177. A few may ask for a letter of explanation, but considering that it is 6+ years old, I doubt it.
I would be ready to explain it nonetheless.