Kingston Park is signed as a Park & Ride from the A1/A696 junction. The station itself is by the giant Tesco, just be careful that you get the right platform, they are split by the level crossing.
If your driver is taking the A19 via the Tyne Tunnel, then Northumberland Park station is your best bet, just off the A19/A186/A191 junction.
If you're willing to stomach the bus instead, there's also a new Park & Ride at "Great Park", just off the A1/A1056/B1318 junction, which will be easier to find by road than the metro, providing your driver doesn't want to stray far from the A1.
Avoid Newcastle airport like the plague. There's a substantial charge just for using the drop off area. If you want to go out that way, Callerton Parkway station is far easier. In fact I suspect that Callerton Parkway is the easiest to find, as it is right by the A696 (no hunting round housing estates), last junction on the dual carriageway before the airport. It's 2-3 miles away from the A1, but the A696 is a fast dual carriageway.