Voici ce que j’ai trouvé.

https://www.thelocal.fr/20181102/french-word-of-the-day-bof

It’s a French classic you need to know. Why do I need to know bof?

If you’re feeling demotivated, indifferent, or want to engage in the
traditional French pastime of avoiding being positive (being honest)
about things, this is a need to know word.

Plus it’s a French classic, right up there with pfff, exaggerated
shrugging and oh la la.

What does it mean?

Historically it’s thought that this word might be linked to the
acronym of Bœuf, Œuf, Fromage. All three foods were rationed during
the German war-time occupation in France and black marketeers became
known as BOFs. Overtime bof has lost this unscrupulous association
and come to mean something quite different.

Bof is a spoken interjection that translates more as a feeling of
disinterest or mild unhappiness than an actual word.

It’s nearly always used as an indifferent or slightly negative
response to a question, for example, – Que penses-tu de ce film? –
Bof. Pas terrible. (What did you think of the film? – Whatever. It
wasn’t terrible.)

Similarly bof could also be the response to ‘Don’t you think the film
is great?’ (Tu trouves pas que ce film est génial?) or ‘Do you want to
go to the cinema? (Ca te dit d’aller au cinéma?), meaning an apathetic
‘not really’ in both cases.

It could also be a slightly depressing reply to ça va? meaning ‘not
great’, ‘ok’, or ‘meh’.

Considering that a normal reply would be ‘fine’ or ‘good thanks’
(bien, merci) saying you are just ‘alright’, ‘ok’ or bof actually
implies that you feeling a bit miserable.

Finally, if you’re going to use this classic French sound you might as
well go the whole hog and Frenchify your gestures too; bof is often
said with an indifferent expression and dismissive shrug of the
shoulders.

How can I use bof?

• Est-ce que tu as faim? – Bof…

• Are you hungry? – Not really…

• Tu t-amuses bien? – Bof.

• Are you having a good time? – Meh.

Un article français plus court affirme aussi cette étymologie :

https://www.gqmagazine.fr/mode/style-academie/articles/quest-ce-qui-differencie-le-bof-du-beauf-/23915

L’expression, qui n’est pas une bête contraction de “beau-frère”,
révèle un acronyme plus recherché : Beurre Œufs Fromage. Au temps de
la Seconde Guerre mondiale, les restrictions alimentaires font le lit
du marché noir. Sous l’enseigne “Beurre Œufs Fromage”, les épiciers
les moins scrupuleux se font du blé sur le dos des pauvres. L’insulte
désigne depuis lors une personne sans éthique ni scrupule.

D’après languefrancaise.net, la plus ancienne référence de bof avec cette acception date de 1967 L’argot chez les mineurs délinquants

Pour ce qui est de l’étymologie, il s’agit d’une onomatopée reproduisant le bruit de l’air de joues gonflées sortant par les lèvres se desserrant, similaire à pfft qui existe aussi en anglais.

Grevisse confirme le caractère onomatopéique de bof (LBU, 16e éd., 2016, p231).