Others have explained the derivation of the word 'Rake'.
Significantly, in the case of these two railway stations, there is a simple explanation. They are sited on or very near to roads which already existed when these relatively new stations were built. Pedestrian access to Eastham Rake station is from a road called 'Eastham Rake', and Bromborough Rake station is at the end of the road [in Bromborough] called 'The Rake'.
As for why these roads were called 'Rake', rather than 'Avenue', 'Street' etc, the
Cambridge Dictionary includes the following definition:
rake noun [ C ] (MAN)
especially one who is rich or with a high social position, who lives in an immoral way, especially having sex with a lot of women
Perhaps these were (or still are) the areas inhabited by such people?