The baptism entry lies in Banbury Registration Area CD Rom 1.
Valentine is apparently from Grimsbury which is like Clifton in a way, except this suburb is close to Banbury itself.
I have to point out that I found a William being baptised to a Thomas (a tailor) in 1707 which I found after I posted but since you have a Valentine in the tree already I'd lean more towards the first William but I'm always cautious to keep the second one in mind.
Looking on the Marriage Index there is one entry of a Valentine GARDNER marrying an Elizabeth in Wroxton 1702 and seems to fit our William of 1702 despite the early birth. But this is where it gets confusing....and it seems there may be 2 Valentine GARDNERs....
There doesn't seem to be anymore baptisms for Valentine GARDNER after or before 1702 in Banbury. Which had me think it was this Valentine and Elizabeth because in Wroxton they don't have any children until Thomas in early 1704. Unfortunately in 1705 Valentine and Elizabeth have a child baptised William which throws doubt on this as I cannot find a burial previously for a William in either parish.
Wroxton is on Banbury CD Rom 2
A Robert GARDNER also names his child Valentine in Banbury so it seems not as uncommon as we think in the area.
I just looked in Cropredy a parish further north and it is full of GARDENERs (Banbury CD Rom 2) and yes, there is a William being baptised in 1702/3 to a William/Elizabeth.
I tend to think it may be wise to get Banbury CD Rom 1 and 2 just to see if you can sort out the various GARDNERs there. But this is just an suggestion.