"Strong use of the word" or "...phrase" seems to have a wide variety of meanings, some of which are contradictory: sometimes it can refer to a narrow or particular use, but often it refers to a questionable or inaccurate use. It can also refer to the use of bold or striking language, or to frequent use of a word.
In the absence of definition I'll look at some examples.
In the Working Papers of the National Commission on Reform of Federal Criminal Laws Relating to the Study Draft of the New Federal Criminal Code: Relating to Chapters 1-13 of the Study Draft of a new Federal Criminal Code (U.S. Government Printing Office, 1970) we find a debate over whether the word "conduct" can include involuntary actions, with "strong use" referring to a narrow or more precise definition (this may relate to the phrase a fortiori meaning "with greater reason" or "as a stronger case"):
At least the strong use of the word 'conduct' contains some notion of
a person's conducting himself in a certain way, exercising control
over himself, some element of volition.
On the other hand, many more recent sources use the phrase to mean questionable - a wide or less precise use. From the BBC in 2014:
Dr Goodenough said she just wants participants' best guess, no matter how tentative the speculation.
"It's a strong use of the word 'estimate'," she said.
Similar in meaning, from the Guardian in 2003, a discussion of a phrase in a book review:
"a woman in a yellow hat in front of the lemons on display in such a way that the hat becomes a lemon". (A rather strong use of the word "becomes" there, perhaps.)
Another sense is to mean an emphasised or striking word or phrase. In the following, from a lighthearted report (Diary) in a British news website in 2017, it appears to refer to a bold or striking use of a word:
What was supposed to be a controlled bomb demonstration went horribly, if somewhat amusingly, wrong and two men had to be taken to a nearby hospital after being injured by the device.
Attendees of the re-enactment obviously took to social media, with one user rather helpfully noting that “someone legit just got blown up” at the event. Which is, when you think about it, quite a succinct and helpful statement given the situation. Strong use of the word ‘legit’, despite the fact that they had more than enough available Twitter characters to go with ‘legitimate’.
The Daily Mail in 2012 uses "makes strong use of" to refer to repeated and emphasised use:
Visit Suffolk launched a new promotional drive last month which made strong use of the word ‘curious’ as a way of highlighting the attractions of the East Anglian region.
Twitter users were encouraged to tweet about the county via hashtags such as #proudtobecurious and #curiouscounty.