The backlash has come in response to a press release issued by the MoJ last week, in which it asked for suggestions for what to call a £235m "mega prison", the construction of which began last month.
"Residents are encouraged to embrace the history and culture of Wellingborough when sending in their suggestions," it stated.
The prison is being built in the Northamptonshire town on the site of an ageing and outdated jail that was shut down in 2012.
Clinky McClinkface, Cellingborough, HMP Spiceworld; Spicy McSpice Face, Prisoney McPrisonface, Locky McLockup, and Jaily McJailface are among the suggestions doing the rounds on social media.
HMP Clinky McClinkface?https://t.co/AiVQEY9iQ6— Alec Muffett (@AlecMuffett) October 17, 2019
They echo the embarrassing controversy in 2016, when Boaty McBoatface was the top choice in a poll of names suggested by the public for the UK's new polar research ship. The result was subsequently ignored, and the ship named the RRS Sir David Attenborough instead.
Rheanne Scott, a former strategy advisor at the Youth Justice Board, tweeted: "Can't help but think this is ill-advised. Can we all remember the 'Boaty McBoat Face' debacle?"
And Frances Crook, chief executive of the Howard League for Penal Reform, commented: "Putting the naming of a new prison to a public vote is patronising and insensitive. Building the prison in the first place is a huge mistake."
In her view, the money being spent "would be better invested in the community – in the NHS, drug and alcohol services, education and local employment".
Wellingborough prison: Public invited to pick the new jail's name https://t.co/Hs8D0xxIJu Are the #BBC and MOJ taking the piss or what?! It’s bad enough that £235 million is spent on a prison, but then to ask the public to name it?! ??@RandolphTrent @godwontlikethis @HanboMillar— Madame Butterfly (@0010011Effect) October 20, 2019
The category C prison, which will have capacity for nearly 1,700 prisoners, is set to open in 2021.
Giving people the chance to name it is part of the MoJ's wider approach to community engagement.
It will be the first time that the public will have named a prison in England, and comes after a similar exercise in 2016, where a prison in Wrexham – the biggest in England and Wales – was named HMP Berwyn.
In the case of the new prison, another reason for consulting on a name is that its predecessor on the same site was called HMP Wellingborough.
"The names will be narrowed down to a shortlist of potential options, with a panel of local representatives deciding on the final winner," the MoJ release stated.
To guard against another Boaty McBoatface-type scenario, "the name must then be formally agreed by Her Majesty The Queen, the Prisons Minister, Lucy Frazer QC MP and Secretary of State for Justice, Robert Buckland QC MP," it added.
Click here to subscribe to the FREE public sector bulletin to receive dedicated public sector news, features and comment straight to your inbox.
Make sure you register for the site to access more than one story per month.
To submit a news, comment, case study or analysis idea for the public sector bulletin, email Ian.Griggs@haymarket.com