N/W : H1(eth0)--- LAN1
(eth1)--- LAN1

The below route stops my LAN connection .

# ip route show
192.168.1.0/24 dev eth0 proto kernel scope link src 192.168.1.59
192.168.1.0/24 dev eth1 proto kernel scope link src 192.168.1.2
127.0.0.0/8 dev lo scope link
default via 192.168.1.254 dev eth1 proto static

# ifconfig eth0 down

Then
# ip route show
192.168.1.0/24 dev eth2 proto kernel scope link src 192.168.1.2
127.0.0.0/8 dev lo scope link
default via 192.168.1.254 dev eth2 proto static

How do I make this route take precedence over eth0 while keeping eth0 for networking to my lan ?

Use the route command to set the metric on duplicate routes. The route with the lower metric is used for forwarding.