The SussexRoyal Instagram account shared an image of Meghan, who is American, holding her son's heel.
In the caption, the account paid tribute to "all mothers today, past, present, mothers-to-be, and those lost but forever remembered".
While Mother's Day is in March in the UK, it was marked in the US on Sunday.
It was also celebrated in Canada, New Zealand, Australia, South Africa, Kenya and Japan.
The photo was accompanied by the poem Lands, by writer Nayyirah Waheed.
Archie Harrison Mountbatten-Windsor was born weighing 7lb 3oz on Monday.
The Duke and Duchess of Sussex's first son is seventh in line to the throne, behind the Prince of Wales, the Duke of Cambridge and his children - Prince George, Princess Charlotte and Prince Louis - and Prince Harry.
He is the Queen's eighth great-grandchild.