What particular wallet? It's from 2010 so I assume it is a Bitcoin full node.
The wallet.dat contains your mnemonic code that was used to make that one pair of keys. So,
There will be no issue in generating new keys wherever computer you want to transfer the hard drive or wallet aside from the compatibility (if there's any)
You are right about transfering the wallet.dat to a new computer, however, a wallet.dat from 2010 does not contain a mnemonic code.
Nowadays, a default wallet generated with core is HD, it still doesn't have a mnemonic phrase. Back in 2010, bitcoin core wallets were defenatly not HD, they just generated random private keys. These private keys were stored in the (hopefully encrypted) wallet.dat. This is why you had to backup your wallet at least every 100 transactions, since default, 100 private keys were pre-generated (IIRC).
@OP: like many people have already said: you can just copy your wallet.dat to a safe place, then move a COPY of the copy to a new pc... You don't have to wait for your bitcoin core wallet to sync tough: you can just export the private keys and import them into a sync'ed wallet (or an SPV wallet like electrum).
A last option would be to use db_dump or
https://github.com/jackjack-jj/pywallet to export the keys directly from the wallet.dat, this will save you a lot of time and diskspace
Last remark: since you're talking about a 8 year old wallet, there might be compatibility issues with modern versions of core (i really don't know), that's why it's important to keep at least 1 master copy of your wallet.dat in a safe place. I think pywallet is incompatible with more recent versions of the wallet, so it should probably work on really old versions.
Good luck!