Old Roan in Merseyside is similarly named after a defunct pub. I'm not sure if the name also attaches to the area, but if not 'Aintree North' would be better.
The road signs call it Aintree Village, but in the 1980s when I lived there it was known as "the Old Roan" (always with the definite article) after the pub and station. Most of the housing in that area is 1930s, so no doubt there was nothing but the pub to name the station after when it opened. The other side of the railway bridge was Netherton.
The term "Aintree" was more used for the area of primarily terraced housing the other side of the racecourse (near the current Aintree station, once called Aintree Sefton Arms I believe, after a pub that no longer exists), but further from Fazakerley and Orrell Park stations. Though it did also come up for the part of that settlement away from the station. Looking at a Google map showing the name of businesses, it seems Old Roan comes up roughly west of Eton Drive, while Aintree is used for places further towards the racecourse.
Though in the end it's a "local station for local people" not a destination, so it could, as someone said, just be called "the station" for all it matters. In terms of renaming it you could call the two Aintree Village and Aintree Racecourse*, but I don't see a massive point in bothering as neither is fundamentally inaccurate.
* The Cheshire Lines Committee station actually called that was further from the racecourse and would have probably been better called Walton Vale or Melling Road, apart from that it was used racedays only!
