What effect will the time of year a student starts their flight training have on the difficulty of their training?
Different latitudes will have different, or at least more extreme, effects so please note if an effect is only valid for a specific geographic area.
Pretty much the only thing that is going to change season to season is the weather. Depending on where you are, there may be more aircraft traffic around a particular time of year, but I can't answer that for sure.
Obviously, the further from the equator, the more extreme the seasons. If you are close to the equator, it probably doesn't matter what time of year you start, since the weather is pretty much the same year-round.
If you are not at the equator, you need to consider the pros and cons of each weather season. Additionally, at extremes from the equator (near the poles), summer and winter months have extremely long days and long nights, respectively, which can make night and day flying difficult.
Depending on your latitude, other weather conditions may be more prevelant at a certain time of year (hurricanes/cyclones, high winds, tornados, sudden storms). This paper discusses wind differences based on season and latitude. In particular, page 7 of the document shows that summer months had slightly stronger winds at southern latitudes (southern hemisphere), and winter months had significantly stronger winds at northern latitudes. Some geographical areas may have changes in prevailing wind directions, which could also affect training depending on routes and runways too.
TL;DR: It's mainly a weather concern, which can affect your comfort (temperature inside, possibly air turbulence), aircraft performance, as well as grounding weather. Additionally, the most obvious concern is also length of the day - summer months will have more daylight, a particular concern if you can only fly in VFR conditions/do not hold a night rating.
As SSumner says, the weather is your factor for season difficulty in flight training, and in large part these weather differences can be related to geographical location.
In the North US in winter time you have ice threats, for flight training ice is a no-go situation (most trainers are not approved for ice nor do they have a/ice capabilities). For days with no ice, there is often an infinite plane of stratocumulus clouds that you need an IFR clearance to get above. This can still be a go situation for maneuvers but makes ground-based navigation impossible to practice. You also will typically encounter very strong winds as the weather clears behind mid-latitude cyclones.
In the South during the winter you have sporadic freezing rain and icing events, but without the frequency of the North. Your big threat instead are the spring and summer thunderstorm season. On many days, if you respect the AIMs idea of tstorm avoidance, you might as well stay on the ground, because it just doesn't exist.
In the Florida You will have thunderstorms like clockwork daily, all year long. You can fly before and after these storms and the sea breeze that drives these storms makes the timing and position fairly reliable to be avoided.
In the desert southwest you are dealing with high density altitudes due to the elevation and the heat. This can make climb performance problematic but this can be remedied with a capable airplane. This region can also be problematic for instrument training because you'll never get to log any actual instrument time.
In Texas north though Nebraska you can sometimes get intense low-level jets bringing moisture north out of the gulf. This can result in very strong winds at the surface and gusty conditions in lowest few thousand feet, and this can be problematic for training.
In the lee of mountains with strong winds aloft (e.g. New Mexico north along the front range of the Rockies) you can experience turbulence, mountain waves and rotor clouds. These can predicted by the winds aloft and can be avoided but may present days when training is best not conducted.
