If a full featured HTTP proxy is required, the tinyproxy package is an ideal solution for you as a larger proxy might be unavailable in this environment.

1. install software packages:

   opkg update
   opkg install tinyproxy luci-app-tinyproxy
   

2. configure tinyproxy:

   uci set tinyproxy.@tinyproxy[0].enabled=1
   uci commit
   /etc/init.d/tinyproxy enable
   /etc/init.d/tinyproxy restart
   

3. configure transparent proxy redirection:

   uci add firewall redirect
   uci set firewall.@redirect[0].name='Transparent Proxy Redirect'
   uci set firewall.@redirect[0].src=lan
   uci set firewall.@redirect[0].proto=tcp
   uci set firewall.@redirect[0].dest_port=8888
   uci set firewall.@redirect[0].src_dport=80
   uci set firewall.@redirect[0].src_dip='!192.168.1.1'
   uci set firewall.@redirect[0].dest_ip=192.168.1.1
   uci commit firewall
   /etc/init.d/firewall restart
   

Note that the firewall.@redirect[0].src_dip=!192.168.1.1 option is important, if you missed this option you may not connect to LuCI. I can't find this option in the LuCI "Network ⇒ Firewall ⇒ Traffic Redirection" page, so be careful if you're using LuCI.

Note also that by default tinyproxy does not allow connections from other hosts so you will need to enable this. One way is to comment out the "Allow" line from the config.