In order to get a newer Safari browser, you'd have to get a later OS X system upgrade.
However the old MacBook/Pro you have probably cannot run a newest supported OS X
that has a recently updated Safari browser.
The best choice for Snow Leopard 10.6.8 would be to get and use the Firefox v 40. browser
from Mozilla, direct. It can import your bookmarks and also be set to run as a default to the
OS X by choosing that option in the old Safari preferences to select another as default. So
that means any demand for a browser would then launch Firefox instead of outdated Safari.
If you are in the US, the Mozilla Firefox link may look like this, in other regions a different URL
would be available; sometimes by just clicking on the US english page address:
https://www.mozilla.org/en-US/firefox/new/
Anyway, I'd be reluctant to attempt to buy & install Lion 10.7.5 even if it could work, as
some older applications may need to be upgraded, in some cases. Among other issues.
Your model MacBook/Pro could see an upgrade in memory, buy purchasing additional RAM
& replacement of existing chips to larger capacity exact spec chips from a reputable vendor.
PS: The first CoreDuo series has limits to upgradability, compared to Core2Duo processors.
In any event...
Good luck & happy computing! 🙂