Rhodri Philipps, 50, the 4th Viscount St Davids, made the threat in November, just days after Miller won a legal case requiring the government to consult parliament before beginning Britain's withdrawal from the European Union.
"£5,000 (5,600 euros, $6,400) for the first person to 'accidentally' run over this bloody troublesome first generation immigrant," he wrote.
He described Miller, who was born in what was then British Guiana, as a "boat jumper" and added: "If this is what we should expect from immigrants, send them back to their stinking jungles."
Judge Emma Arbuthnot said the comments by Philipps, who is also known as Lord St Davids, were racially aggravated.
In a statement read to the court in London on Monday, Miller said the comments left her "very scared for the safety of herself and her family".
Prosecuting lawyer Philip Stott said: "She took the threat seriously, and it contributed to her employing professional security for her protection."
Philipps, who lives in the exclusive London district of Knightsbridge, accepted writing the posts but said they were intended to be humorous and were visible only to his Facebook friends.
Judge Arbuthnot ordered the recently bankrupt Philipps to pay £500 compensation.
"You are not motivated by love of country, but by your hatred of anybody who has different views to yours," she told him.
"You show this hatred by publicly directing abusive threats at others, which is a criminal offence in this multi-racial society we are lucky enough to live in."
Miller was subjected to torrents of abuse for her legal challenge, which critics took as an attempt to block Brexit. The government began the process of leaving the EU in March after a referendum last year.
Philipps was also convicted for another post in which he offered money to "carve... into pieces" an immigrant reported to have turned down a five-bedroom council house for his family of eight children.
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