Learn Hardcore Finnish: Ystäväni pitää suomenkielistä blogia, jossa hän kertoo arjestaan. - My friend has a Finnish-language blog where she/he talks about everyday life.

Both are correct.
Using just the suffix (Ystäväni) is a bit more compact and slightly more formal/neutral. Using minun ystäväni adds some emphasis to my (as opposed to someone else’s friend), and is very common in spoken language (often as mun ystävä).

Why is it pitää blogia and not something like “on blogi” for “has a blog”?

In this sentence pitää blogia means “to run / maintain a blog”, not just “to have one”.

pitää blogia ≈ to keep a blog, to run a blog regularly
hänellä on blogi = he/she has a blog (just stating possession)

So:

Ystäväni pitää suomenkielistä blogia
= My friend runs a Finnish‑language blog.
Ystävälläni on blogi
= My friend has a blog.

The verb pitää here is “to keep / maintain”, not the “to like” meaning.

Why is it blogia (partitive) and not blogin or just blogi?

Blogia is the partitive singular of blogi.

With the meaning “to keep / maintain something as an ongoing activity”, the verb pitää normally takes a partitive object:

pitää kauppaa = to run a shop
pitää ravintolaa = to run a restaurant
pitää blogia = to run a blog

So blogia is partitive because of the verb pattern: pitää + partitive in this sense.
Forms like pitää blogin are not used in this meaning.

What does suomenkielistä mean, and how is it formed?

Suomenkielistä means “Finnish‑language” or “in Finnish” (as an adjective describing the blog).

Formation:

suomi = the Finnish language
suomen = genitive of suomi (“of Finnish”)
kieli = language
-nen → -llinen / -kielinen type adjective: kielinen = “‑language, -tongued”

So: suomen + kielinen → suomenkielinen = “Finnish‑language”.

In the sentence we see suomenkielistä, which is the partitive singular form of the adjective suomenkielinen, matching blogia.

Compare:

suomalainen blogi = a Finnish blog (from Finland / by a Finn)
suomenkielinen blogi = a blog in Finnish (language)

Why is suomenkielistä also in the partitive case?

In Finnish, adjectives agree with the nouns they modify in:

number (singular/plural)
case

The noun here is blogia (partitive singular), so the adjective must also be partitive singular:

nominative: suomenkielinen blogi
partitive: suomenkielistä blogia

So suomenkielistä is partitive just because blogia is partitive.

What does jossa mean here, and what does it refer to?

Jossa means “in which”.

It is the relative pronoun joka (“who/which/that”) in the inessive case (-ssa, “in”):

joka (which) → jossa (in which)

In the sentence, jossa refers back to blogia / blogi:

…blogia, jossa hän kertoo arjestaan.
= “…a blog, in which he/she talks about his/her everyday life.”

So it’s like saying “a Finnish‑language blog where she talks about her daily life”, but more literally “in which she tells…”.

Why is it jossa and not just joka or missä?

You can’t use bare joka here, because the meaning is “in which (blog)”—you need the local case:

joka = who/which/that (basic form)
jossa = in which (inessive)
missä = where (question/relative adverb)

Jossa is the correct relative form that links back to a specific noun (blogi).
Missä is more independent and would usually start a new clause/question (“Where…?”), not refer directly back to blogi as a relative pronoun.

So:

blogi, jossa… = the blog in which… (correct here)
Missä hän kertoo arjestaan? = Where does he/she talk about everyday life? (question, different structure)

What is arjestaan made of, and what does it literally mean?

Arjestaan means “about his/her everyday life”.

Morphology:

arki = everyday life, daily routine
arjesta = from / about everyday life (elative case: -sta/-stä)
arjestaan = arjesta + -n (3rd person singular possessive suffix = his/her)

So:

arjesta = from / about everyday life
arjestaan = from / about his/her everyday life

Literally: from his/her everyday life → in English: “about his/her daily life”.

Why is the ending on arjestaan -sta (elative)? Why not a different case?

The verb kertoa (“to tell”) typically uses the elative case for the thing you are talking about:

kertoa jostakin = to tell about something
- jostakin = from/about something (elative of joku: joku → jostakin)

So:

kertoa arjesta = to tell about everyday life
kertoa arjestaan = to tell about his/her everyday life

Using arkea (partitive) would change the grammar and sound wrong here; the standard pattern after kertoa is kertoa + elative.

Could the hän be omitted in jossa hän kertoo arjestaan?

Yes, grammatically you can omit the subject pronoun in Finnish if it’s clear from context:

…, jossa hän kertoo arjestaan.
…, jossa kertoo arjestaan.

Both are possible.

In written standard Finnish, hän is often kept for clarity and formality.
In casual speech or more relaxed writing, dropping it is common when the subject is obvious from the previous clause (Ystäväni).

Is the comma before jossa necessary?

Yes, in standard written Finnish you normally put a comma before a relative clause starting with joka/jossa/josta…:

Ystäväni pitää suomenkielistä blogia, jossa hän kertoo arjestaan.

The part jossa hän kertoo arjestaan is a separate clause describing the blog, so it’s set off by a comma, like in English:

“My friend runs a Finnish‑language blog, in which she talks about her daily life.”

Can the word order change, for example to Ystäväni suomenkielistä blogia pitää?

Yes, Finnish word order is relatively flexible, but neutral word order here is:

Ystäväni pitää suomenkielistä blogia, jossa hän kertoo arjestaan.

Other orders are possible, but they change emphasis:

Ystäväni suomenkielistä blogia pitää
– puts extra focus on pitää (the keeping/running), sounds marked/poetic.
Ystäväni pitää blogia suomenkielistä
– sounds odd/wrong; the adjective should normally be placed before the noun it modifies.

So for normal, clear Finnish, keep:

[Ystäväni] [pitää] [suomenkielistä blogia]…