Margaret McGarry is right, Hamilton Road as exists today is a fairly new street. Old Motherwell folk still call the part between Avon Street and Malcolm Street "Jack's Brae".
Here is a description of the area in the 1840s by J A King from his book Motherwell 70 Years ago and Now, published in 1910.
"Turning down Clyde Street, one came to Paterson's pond, which lay where the Town Hall is Built. This pond was at the entrance to High Motherwell Farm.............continuing down the road one came upon the entrance to Low Motherwell Farm.......on leaving the road to Low Motherwell, one went down Kether's Brae and across Kether's Burn, now miscalled the Soo Burn. The road then continued much as it is now, with plantations on each side down to the pay bridge (Clyde bridge)".
That description now reads - Turning into Hamilton Road (from Brandon St or Muir St) past the old Town Hall, all the way down to The Loaning (on the right), turn left into Tinkers Lane, across Airbles Road into Airbles Farm Road, all the way down to where the old Clyde Bridge used to be.
King also describes Jack's Brae (where Dr Jack had his house) and further down, Forgie's or Frame's Brae.
The only mention of "Hamilton Road" he makes is in the present (1910).
He mentions many old Motherwell families, perhaps yours is there too?
Here is a picture of Hamilton Road (looking down Jack's Brae) taken about 1910. It would have looked nothing like this in 1861.
Also a picture of the old Clyde Bridge as a ruin before it was demolished about 40 years ago.