Just because a restaurant is BYOB (bring your own bottle), it does not mean that you have to bring something that is cheap and not flavorful. There are plenty of options when dining out at restaurants that allow you to bring your own alcohol, be it beer or wine. Many people tend to associate BYOB restaurants with cheaper bottles of wine and six packs of not too expensive beer. That is a common misconception. In a lot of places it's because there happens to be wine stores which profit by selling fast to people looking to buy something quick and easy. However, if you plan it out a bit, you can certainly find some really amazing wine, beers, and other alcoholic beverages to bring to your next dinner at a BYOB.
This list will give you a few simple suggestions to consider.
If you are going somewhere that serves food that you'd love to pair with beer (maybe Chinese or something similar) then you should definitely check out some really great craft beer. You don't have to just grab a sixer of the local mass produced beer product. There are some amazing craft brews that you can find in most local supermarkets and other locations that sell beer. Even if you don't live near some of the smaller beer stores, you will have have lots of options. Even the mass producers of beer are now making thing such as IPA style beers and stouts that have a robust flavor.
If you are eating something such as Greek or maybe fast casual seafood, and you are searching for a white wine that is not chardonnay, then think about Sauvignon Blanc. This is a beautiful wine that would go perfectly with something such as Greek food or seafood. If you're looking for something that is not as heavy as Chardonnay, then definitely check it out. You can get some really affordable varieties and most places will sell it chilled so you don't have to worry about drinking warm white wine.
One really amazing wine to check out is the varietal known as Sauternes wine. This is particularity great if you're looking for wine that will stand up to strong flavors. It is perfect for Italian food, especially cheese dishes such as fettuccine Alfredo and eggplant parm. You can get a really great sauternes blend that will be amazing chilled and will compliment any food dish that you've chosen.