Learn Hardcore Norwegian: Bestemor sier at hun beundrer barnebarnas tålmodighet. - Grandma says that she admires the grandchildren’s patience.

In Norwegian, kinship terms like Bestemor (Grandma) are often used like names, especially when referring to your own grandmother.
So Bestemor here works like a proper name (like Anna) and doesn’t need an article.
If you say en bestemor, it just means a grandmother (in general), and bestemoren means the grandmother (a specific one, but not used as her “name”).

Why is Bestemor capitalized?

It’s capitalized because it’s being used like a proper name, similar to Grandma in English when you use it as someone’s “title name” (e.g. I’ll visit Grandma).
If you were talking about grandmothers in general, you would usually write bestemor with a lowercase b.

What does at do in sier at hun beundrer …? Is it like English “that”?

Yes, at is the conjunction that introducing a subordinate clause.
Bestemor sier at hun beundrer barnebarnas tålmodighet = Grandma says *that she admires the grandchildren’s patience.
In the *at-clause, Norwegian uses normal subject–verb–object order: hun beundrer barnebarnas tålmodighet (not verb-second like in main clauses).

Could you drop at and just say Bestemor sier hun beundrer barnebarnas tålmodighet?

Yes, that is possible in everyday spoken Norwegian and is understandable: Bestemor sier hun beundrer barnebarnas tålmodighet.
However, including at is more neutral and standard, especially in writing, so learners are usually advised to keep at.

Why is it hun and not henne in at hun beundrer …?

Hun is the subject form (she), while henne is the object form (her).
In the clause hun beundrer barnebarnas tålmodighet, she is doing the admiring, so hun must be the subject.
You would use henne as an object, e.g. Bestemor sier at vi beundrer henne (Grandma says that we admire her).

Does beundrer change with the subject, like English “admires / admire”?

No, Norwegian present tense verbs have only one form for all persons.
Jeg beundrer, du beundrer, han/hun beundrer, vi beundrer, de beundrer – all use beundrer.
So you don’t add an -s for third person singular the way English does.

What exactly is going on in barnebarnas? Why the -s at the end?

Barnebarnas is a genitive form meaning the grandchildren’s.
Breakdown: barnebarn (grandchild / grandchildren) → barnebarna (the grandchildren) → barnebarnas (the grandchildren’s).
Norwegian often uses -s on the end of a word to mark possession: barnebarnas tålmodighet = the patience of the grandchildren / the grandchildren’s patience.

Why is the plural of barnebarn not barnebarn → barnebarn + er?

Barnebarn is one of those neuter nouns whose plural is identical to the singular: et barnebarn (one grandchild), to barnebarn (two grandchildren).
To make it definite plural, you add -a: barnebarna = the grandchildren.
Then the genitive -s attaches to that: barnebarnas = the grandchildren’s.

Could you also say tålmodighet til barnebarna instead of barnebarnas tålmodighet?

You could say tålmodigheten til barnebarna, but that usually means the patience *towards the grandchildren (who is patient *with them).
Barnebarnas tålmodighet clearly means the grandchildren’s patience (the patience that the grandchildren have).
So the -s genitive is the natural choice for possession.

What is tålmodighet, and how is it related to tålmodig?

Tålmodig is the adjective patient, and tålmodighet is the abstract noun patience.
It’s usually treated as a mass noun: hun har mye tålmodighet (she has a lot of patience).
You can make it definite if you’re talking about specific patience: tålmodigheten hennes (her patience).

Is the word order hun beundrer barnebarnas tålmodighet fixed, or can I move things around?

In this subordinate clause after at, the normal order is subject – verb – object – other elements.
So hun beundrer barnebarnas tålmodighet is correct; something like hun barnebarnas tålmodighet beundrer would be ungrammatical in standard Norwegian.
Norwegian does allow some topicalization, but learners should stick to the straightforward S–V–O order inside at-clauses.