The root of al-Badīʿ and bidʿah

One of the beautiful Names of Allāh subhanahu wa taala - البَديعُ
The One who brings into existence without prior example, prior precedence

This Name comes from the Arabic root letters
ب د ع
which carry the meaning of originating something new without any prior example.

In Arabic morphology, words derived from the same root generally keep a shared core meaning.

As an example - the word بِدْعَةٌ
It refers linguistically to something newly originated without precedent.
This includes anything newly introduced, whether good or bad.

And in its shariah definition, it refers to introducing something into the religion that has no basis in the Shariah.

Another example of the word sharing the same root letters - بِدْعٌ

It is related to بِدْعَة, but more general in meaning, referring broadly to anything unprecedented, whereas بِدْعَة refers to a specific newly introduced matter.

In the Qurʾān:

قُلْ مَا كُنتُ بِدْعًا مِّنَ الرُّسُلِ
“Say: I am not something unprecedented among the messengers…”

[Sūrat al-Aḥqāf: 9]

So in all these words the same shared meaning remains -
to originate something entirely new and unprecedented.