Those are the specs mid-range Chromebooks tend to have. Besides swapping your hard drive for an SSD, look into installing a really lightweight distro that's specifically geared towards web surfing.
This is a Ubuntu derivative inspired by the Chrome OS. It places the open source Chromium web browser front and centre with . The minimum requirements for this distro are 512MB RAM (1GB preferred), 1GHZ PAE/SSE2 capable Intel/AMD processor.
Chromixium combines the elegant simplicity of the Chromebook with the flexibility and stability of Ubuntu’s Long Term Support release. Chromixium puts the web front and center of the user experience. Web and Chrome apps work straight out of the browser to connect you to all your personal, work and education networks. Sign into Chromium to sync all your apps and bookmarks. When you are offline or when you need more power, you can install any number of applications for work or play, including LibreOffice, Skype, Steam and a whole lot more. Security updates are installed seamlessly and effortlessly in the background and will be supplied until 2019. You can install Chromixium in place of any existing operating system, or alongside Windows or Linux.
This is an independent project, also inspired by Chrome OS. However, it is written from scratch, without any parent project. This allows it the flexibility of crafting a fully integrated experience and include their own optimisations.
Solus Operating System is completely free and open source. We leverage fantastic open source software to enrich the end user experience... We believe an operating system should get out of your way, and do its job as well as it possibly can when it is needed.