I haven't really got local knowledge but I drive through Bootle quite often. I googled "Bootle accommodation" and the Throstle's Nest Hotel came up. It's quite well reviewed on Tripadvisor and they do have family rooms. It has it's own website. I thought it was just a pub, but it seems to have been refurbished as a hotel. It's very handy for the football stadia, both Liverpool and Everton. Bootle isn't the smartest part of Liverpool, and this hotel, which used to be known as "One Flew Over The Throstles Nest" is on a main road quite near the tunnel entrance for Wallasey. It's on the Liverpool side of Bootle, on Scotland Road.

You might be able to find a B & B by googling. Alternatively you could stay in Liverpool itself as Bootle isn't that far out. Hope this helps.