Design a Python web scraper that extracts information about books from an online bookstore website and stores the data in a SQLite database.

To design a web scraper in Python, we can use libraries like Beautiful Soup and Requests. We'll also need the sqlite3 library to interact with SQLite databases.

Here's an example of how you could create a simple web scraper to extract book information from an online bookstore website (in this case, I'm using http://books.toscrape.com) and store it in a SQLite database:

```python
import requests
from bs4 import BeautifulSoup
import sqlite3

# Create a connection to the SQLite database
conn = sqlite3.connect('books.db')
c = conn.cursor()

# Create a table for storing book data
c.execute('''CREATE TABLE IF NOT EXISTS books
             (title TEXT, price REAL, rating INTEGER)''')

# Function to get the rating as an integer value
def parse_rating(rating_str):
    if "One" in rating_str:
        return 1
    elif "Two" in rating_str:
        return 2
    elif "Three" in rating_str:
        return 3
    elif "Four" in rating_str:
        return 4
    elif "Five" in rating_str:
        return 5
    else:
        return 0

# Main scraping function
def scrape_books(url):
    response = requests.get(url)
    soup = BeautifulSoup(response.text, 'html.parser')

    # Find all article tags on the page
    articles = soup.find_all('article', class_='product_pod')

    # Extract book data from each article tag
    for article in articles:
        title = article.h3.a['title']
        price = float(article.select_one('.price_color').text[1:])
        rating = parse_rating(article.p['class'][1])

        # Insert the book data into the SQLite database
        c.execute("INSERT INTO books VALUES (?, ?, ?)", (title, price, rating))

    # Check if there's a next page and scrape it
    next_page = soup.select_one('.next > a')
    if next_page:
        next_url = url.rsplit('/', 2)[0] + '/' + next_page['href']
        scrape_books(next_url)

# Start scraping from the first page
scrape_books('http://books.toscrape.com/')

# Commit the changes and close the database connection
conn.commit()
conn.close()
```

This script will create an SQLite database named `books.db` with a table called `books`. It will then scrape book information (title, price, and rating) from each page of the website and insert the data into the `books` table.

Please note that web scraping may be against the terms of service for some websites. Always check a website's "robots.txt" file or terms of service before scraping to ensure you're allowed to access the site programmatically.