Well there's a path that runs parallel to the railway line, which can be accessed from Old Shamblehurst Lane, just a few steps away from the railway station. It terminates at the bridge which carries the Botley to Winchester Road. Not sure I'd describe it as a nice walk though.
Actually, don't think that there's anything really special if you insist on starting at the railway station.
The best walks locally are a little distance from the railway station, so you might need to catch a bus.
Walk 1. The length of Brook Lane and Church lane which essentially runs in a semi-circle around the south of Botley. There are Botley Quays to look at, an ancient parish Church and if you go into the back entrance to Manor Farm Country Park (adjacent to the Church) there's an even older duckpond which is listed in the Domesday book. Besides all that, there are points on that route where you really do feel that you are miles from any sort of civilisation, looking across open fields with hardly any buildings to distract you.
Walk 2. the length of Manor Farm Country park. Starting at the Botley end (by the ancient church), taking in as much of the banks of the RIver Hamble as you can and finishing at the main entrance, Dodwell Lane. The site is oozing history, Manor Farm, Roman Road, medieval shipbuilding, the wreck of the Grace Dieu, the WWII landing craft base (HMS Cricket) and post war Cricket Camp. It was also the site of one of the few V1 impacts locally, during WWII. I always feel the ghosts of all of those various activities whenever I walk through the woods there. Particularly if it is a quiet day and there are few other people around.
Well, they're my two favourite walks anyway, for the reasons I have explained.
Maybe some other contributors will have their own favourites...?
This link might help http://tinyurl.com/326slv7
The route is reasonably well marked, though you will need to follow your nose a little in some places. There are some well defined paths and others which demand some hiking boots a hiking pole and a bit of bog-trotting at this time of year...
The nearest railway station is Botley, rather than Hedge End. To get to the starting point from there 1. Walk into Botley square. 2. Have cooked breakfast in the cafe, you're going to need it! 3. At the crossroads, turn into Church Lane and just keep following that road until it becomes Brook Lane. At that point you can turn left into the Manor Farm Entrance, noticing the old Church and duckpond that I mentioned earlier. 4. The Hants County Council maps of the site should then tell you all you need to know. 5. You have the option of making your walk a circular one, so you finish where you started or you can either exit the park at Pylands Lane and walk up Sundays hill, down the other side and catch a bus to Botley in Heath House Lane, or (least adviseable) you can take the really boggy Tanhouse Lane and leave by the Norman Rodway recreation ground, which is on the same bus-route back into Botley....
Though I doubt the Bus Driver will let you on board if you are too muddy !
I live near Hedge End Railway station and it's a struggle - there are paths, but to say they'd make a nice walk is extending it a bit.
You may be better off getting off at Botley station, which is handier for the walks mentioned above and there is also a pleasant walk going over in the direction of Durley.
I probably wouldn't tackle any without an OS map or at least a local printout.