This would be better answered on Stack Exchange Electrical Engineering.
Also see this question and answers. This is not quite the same and the answers there are OK but less than complete or correct, but it adds to the resource available.
You need a battery charging controller, a PV (solar) panel capable of producing somewhat more than maximum battery voltage and enough energy to operate your circuit and a battery with enough voltage to operate the device and enough capacity to store the energy.
Current drain will vary with total circuit. You can measure this will a milliammeter. I'll assume that 50 mA continuous will be enough. Less or much less may suffice. I have a pro mini clone operating at 20 mA on 5V at present and with due care you may be able to operate one with less or much less current depending on application and what your circuit needs are.
Looking at solar current needs at present, the average solar current needed is about Icircuit x 24/SSH mA where SSH = average sunshine hours per day. eg if you had 4 hours of sunshine per day but want to run the circuit for 24 hours a day the sun must provide (24 hours per day)/(4 hours of sun) = 6 x as much current as the circuit needs on average. So if your circuit needs 50 mA continuous the PV panel must provide 6 x 50 mA = 300 mA for 4 hours per day. You will see below that this is not a trivial amount.
Sunshine hours per day may be found for sites worldwide using the excellent www.gaisma.com The average sunshine hours per day for Iráklion may be found at that link and other locations in Greece can be found here but will probably be similar.
For Iráklion the SSH may be found in the 1st line of the 4th chart.
Average hours/day of equivalent full sunshine = kWh/m^2/day = a peak of 8.53 hours in June and a low of 2.08 hours in December.
I think that is about as sunny a place in Summer as anywhere on earth not in the polar regions. In Winter it gets about the same SSH as in my city. You can calculate PV sizing for peak summer - but if you want year round operation you should use the worst case winter month and add a bit for bad days.
For Iraklion us say 2 SSH per day so the panels needs to provide 24/2 = 12 x winter average load. So per 10 mA of load yo need 120 mA of panel and if the mean load is 50 mA you need 12 x 50 = 300 mA panel output.
Battery voltage needs to be > 5V at all times.
An easy cheap available starting solution is to use a 6B lead acid battery.
These are easyish to charge and provide a relatively stable operating voltage.
Vmax_charge is about 7V so an 8V or 9V PV panel is "about right". 9V is a common voltage for hobby applications. An alternative is to use a 12V panel. this is probably more suitable as they are widely available and cost per energy out is often lower. This wastes MOST of the panel output but still MAY be the cheapest option in many cases. (A 12V x 600 mA panel is rated at about 18V x 0.6A ~= 10 Watts. Actual Arduino load = 50 mA x 5V x 12:1 = 3 Watts)
At 50 mA load you need 24/2 x 50 mA = 600 mA PV output in full sun.
PV panel sizing is not as may be expected. It is easiest to get a panel that makes at least enough voltage and enough current. This is because panel RATED voltages are usually higher than the voltages they are used at. eg 12V system PV panels are usually rated at 18V max power voltage- so an 18W panel provide 18V at 1 A BUT if attached directly to a 12V battery the input to the battery = 12V x 1A = 12 Watts. There are ways around this but at first, just concentrate on panel mA.
So, you "need" a 600 mA panel for a 50 mA load.
Or a 60 mA panel for a 5 mA load.
Or a 6 mA panel for a 0.5 mA load.
Obviously, reducing load current is going to help panel sizing.
So knowing what your application really needs is essential.
Batery is easier.
If you use 6V lead acid then for one day you need.
Hours x load = mAh.
Here load=24 hours. mA = 50 in this example.
mAh = 24 x 50 mA = 1200 mAh = 1.2 Ah. This is the capacity of about the smallest LA battery commonly available. In practice you want to discharge your battery by only a small % of its actual capacity so use a 2.4 Ah or 3.5 Ah or larger. 7Ah is not vastly dearer and will last far longer and will have several days of reserve capacity for bad weather.
Charging can now be discussed - but I'll stop here and see what is said about the above. We can add charging and other detail if it seems useful to do so.
An easy battery to use is 2 x LiIon or LiPo cells in series.