Black's Link is an odd road as it doesn't open up any new building land, the usual reason for building roads in Hong Kong. Instead it was built to enhance Hong Kong's defences against attack from a foreign army landing on the south shore of the island.
The army planned to repel the attack by defending the ridge of hills that form a natural barrier across the centre of the island, and especially the gaps that allowed access to the north shore and Victoria city. However they faced a problem that there weren't the east-west roads that would allow reinforcements to move quickly along the ridge to any gap where they were needed. Here's an extract from page 48 of the book Eastern Fortress, A Military History of Hong Kong, 1840-1970:
When the General Officer Commanding Hong Kong, Major General Wilsone Black, revised the [Defence Scheme of 1897], he noted that a member of the Local Committee had raised much concern about the defence of the south shore, and that the committee had also noted the absence of lateral roads between Victoria Gap, Wanchai Gap, and Wong Nai Chung Gap.
Black proposed a road connecting Wanchai Gap, Middle Gap, and Wong Nai Chung Gap as part of the solution. By the time he left Hong Kong in 1898, the work to build the road had been approved. He described it in his farewell speech to Legco as Acting Governor, Nov. 22, 1898:
Health and pleasure and the wheels of progress, and I may add of bicycles, move on roads, and in my belief a great strengthening of the defence of this Island will take place when the tracing across the pathless barrier of Mount Cameron and Mount Nicholson is broadened into a road. Shortly after arriving in this colony I urged the Government to take this in hand on public grounds, and it is with great pleasure that by an agreement effected on my representation this boon to the pleasure of the colony and to its defence may be expected shortly to begin. The Executive Council has directed that this road be called "General Black's Link." I feel and value the compliment and only ask that the name may be shortened to "The Black Link."
Now we call it "Black's Link".
Thanks to 80sKid for finding Black's speech to Legco.