Are the shrine names supposed to mean something?

Shrines (and their corresponding monks) can be linked to Zelda staff members.

According to this wiki page on gamepedia:

Oman Au is an anagram of Aonuma.
Eiji Aonuma is the lead producer for The Legend of Zelda series.

The Japanese name of the shrine is マ・オーヌの祠 (Ma Ōnu no Hokora), meaning Ma Ōnu Shrine
The Italian name of the shrine is Sacrario di Mau Ona, meaning Mau Ona Shrine.
The French name of the shrine is Sactuaire de Ma'Ohnu, meaning Ma Ohnu Shrine.

These are all anagrams of different translations / romanizations of the name Aonuma.

Quoting from this video of DidYouKnowGaming:

Aonuma is not the only developer that was referenced in this way:
Other monks and shrines are also named after zelda staff:

Zalta Wa Shrine -> Satoru Takizawa ( Art Director )
Sasa Kai Shrine -> Hiroshi Sakasai ( Designer )
Katosa Aug Shrine -> Katsuhisa Sato ( Physics Programmer )

Naming each shrine and monk after a member of the staff was actually suggested by Satoru Takizawa himself.

Based on these examples, not all shrine names are direct anagrams of staff member names, which might make correlating more names to staff members a bit more difficult.

It is still unclear to me whether such a link does indeed exist for every shrine/monk, or that this was only suggested but never fully realized.

Following the sources in the description of the DidYouKnowGaming video, I came back to gamepedia, which has a page with namesakes for many more monks.

Interesting to note is that on the same day I originally posted this answer, a user named TriforceTony added a lot more names to this list!