A point here that is probably VERY important for many here - the point I bolded up above is wrong.OCI will be made upon the basis of your CDN passport because it needs to match since your CDN passport is your travel document not OCI. Hence, it needs to match.
You will need to get a provincial name change done --> get a new citizenship certificate from IRCC --> update your passport --> update OCI.
I applied for a name change in my citizenship application where I split my name and it was approved. It made my application non-routine which took it forever to process but got resolved. The wait was brutal though.
You do NOT need to update your citizenship certificate to change your name in your passport. This is a big deal because you can replace your passport relatively quickly, while changing your citizenship certificate may take quite a long time.*
Documentation is here:
https://www.canada.ca/en/immigration-refugees-citizenship/services/canadian-passports/change-name.html
You only need eg the legal name change document issued by the province to show that you changed your name. That's enough to request a new passport be issued under the new name.
*The fact is, you don't really need to every change the citizenship certificate, but you can if you want to. In almost very instance that will matter in Canada, you can show your passport (new one) with new name to prove you're you and you're a citizen. In the other cases the original citizenship certificate, change of name legal docs, and probably any other identification with a picture (ideally with new name), and that should be enough to convince any institution that you are you, that you have citizenship, and your name is now [new name.]
Note, when I say 'almost every instance', I can't really think of any where that wouldn't be enough.
Sure, change your name in your citizenship certificate if you like - take your time. But no need to wait for that to update your passport.