bugün modern olanın, yarın zamanı geçmiş olacaktı.

I understand this means, "what is modern today will be outdated tomorrow". But why "olanin" ???

another example:

Yüzü güzel olanın huyu da güzel olur .
 Who has a beautiful face has a beautiful character .

http://tureng.com/search/olan%C4%B1n

I see. It´s "Onun zamanı geçti". And then replace onun with the relative clause.

What about the comma after "olanın", even though it´s not a subject?

Teşekkur ederim!

iyi geceler

in this example "olanın" is aproximately equals to "who". 

An expression of relative clause. 

And here the verb is "to be" so "olanın" is used there. 

Another example with that clause but with a different verb accept to be: 

I like books which/that describes other countries. 

Başka ülkeleri "anlatan" kitapları severim.

açıklama: in this example the auxilary verb is "describes"

that´s why "anlatan" is used in turkish translation instead of "olanın" in the first example which you asked.