The invention claimed is:1. A wireless communication device comprising:a wireless communication client configured to (i) perform wireless communication with an access providing device providing access to a predetermined device, and (ii) perform wireless communication with a secondary wireless communication device;a first memory configured to store an evaluation value calculation program, the evaluation value calculation program being configured to calculate a first evaluation value based on at least an estimation of an opportunity for communication between the wireless communication device and the access providing device in a future predetermined time period;a second memory configured to store an evaluation value acquisition program, the evaluation value acquisition program being configured to acquire a second evaluation value for the secondary wireless communication device;a processor configured to (i) read and execute the evaluation value calculation program from the first memory, and (ii) read and execute the evaluation value acquisition program from the second memory;a communication controller configured to control communication between the wireless communication device and the secondary wireless communication device based on a comparison between the first evaluation value and the second evaluation value.2. The wireless communication device according to claim 1, wherein when the communication controller makes a determination based on the comparison between the first evaluation value and the second evaluation value that the secondary wireless communication device has more opportunities to communicate with the access providing device than the wireless communication device does, the communication controller implements control such that information directed to the predetermined device is sent to the secondary wireless communication device.3. The wireless communication device according to claim 1, wherein when the communication controller makes a determination based on the comparison between the first evaluation value and the second evaluation value that the wireless communication device has more opportunities to communicate with the access providing device than the secondary wireless communication device does, the communication controller implements control such that information delivered from the predetermined device is sent to the secondary wireless communication device.4. The wireless communication device according to claim 1, wherein the evaluation value calculation program is configured to calculate the first evaluation value based on at least (i) a communicable time at which the wireless communication device was able to communicate with the access providing device, (ii) a number of communications with the access providing device, or (iii) an amount of communication data in a past predetermined time period.5. The wireless communication device according to claim 1, wherein the evaluation value calculation program is configured to calculate the first evaluation value based on at least (i) a communicable time at which the wireless communication device was able to communicate with the access providing device, (ii) a number of communications with the access providing device, or (iii) an amount of communication data in a past predetermined time period in an area corresponding to a current location of the wireless communication device.6. The wireless communication device according to claim 1, wherein the evaluation value calculation program calculates the first evaluation value based on at least (i) a communicable time at which the wireless communication device can communicate with the access providing device, or (ii) an amount of communication data in a the future predetermined time period estimated based on a scheduled movement path of the wireless communication device.7. The wireless communication device according to claim 1, further comprising an evaluation value transmission program configured to transmit the first evaluation value calculated by the evaluation value calculation program to the secondary wireless communication device, wherein the evaluation value transmission program is configured to incorporate the first evaluation value into a control frame that notifies surroundings of a presence of the wireless communication device.8. The wireless communication device according to claim 7, wherein:a wireless communication system between the wireless communication device and the secondary wireless communication device is a wireless Local Area Network (LAN), andthe evaluation value transmission program incorporates the first evaluation value into an Extended Service Set ID (ESSID) of a beacon.9. The wireless communication device according to claim 8, wherein when the communication controller makes a determination based on the first evaluation value and the second evaluation value that the secondary wireless communication device has more opportunities to communicate with the access providing device than the wireless communication device does, and the wireless communication device retains information directed to the predetermined device, the communication controller establishes a wireless connection to the secondary wireless communication device and implements control such that the information directed to the predetermined device is sent to the secondary wireless communication device.10. The wireless communication device according to claim 8, wherein when the communication controller makes a determination based on the comparison between the first evaluation value and the second evaluation value that the wireless communication device has more opportunities to communicate with the access providing device than the secondary wireless device does, the communication controller functions as an access point, andwhen the communication controller makes a determination based on the comparison between the first evaluation value and the second evaluation value that the wireless communication device has less opportunity to communicate with the access providing device than the secondary wireless communication device does, the communication controller functions as a station.11. A non-transitory computer readable storage medium having a program stored therein for a wireless communication device that causes a computer to execute a process for wireless communication between the wireless communication device and an access providing device configured to provide access to a predetermined device, and with a secondary wireless communication device, the process comprising:calculating a first evaluation value based on at least an estimation of an opportunity for communication between the wireless communication device and the access providing device in a future predetermined time period;acquiring a second evaluation value for the secondary wireless communication device; andcontrolling communication between the wireless communication device and the secondary wireless communication device based on a comparison between the first evaluation value and the second evaluation value.