What is claimed is:1. A server that composes moving thumbnails tailored to users, comprising:a communicator that:acquires information on usage of a moving image content corresponding to a first thumbnail used as a key image of the moving image content in a terminal device that is a distribution destination of the first thumbnail; andacquires a time point during the first thumbnail when the moving image content is requested; anda processor configured to:select a first portion of the moving image content corresponding to the time point;select a second portion of the moving image content based on the acquired information, the second portion of the moving image content being absent from the first thumbnail; andgenerate a new moving image thumbnail in which the second portion of the moving image content is played, and the first portion of the moving image content is played at a middle point or immediately after a start point of the new moving image thumbnail.2. The server according to claim 1, whereinthe communicator acquires, as the information on the usage, information on a user corresponding to the terminal device, andthe processor generates new moving image thumbnails corresponding to each user class classified by user attribute information, based on the acquired information on the user.3. The server according to claim 2, whereinthe communicator acquires, as the information on the user, a social graph indicating a correlation of users in a network environment, andthe processor generates new moving image thumbnails corresponding to each user class classified by the acquired social graph.4. The server according to claim 1, wherein the processor is further configured to:evaluate the first thumbnail by an index value indicating an impact of the first thumbnail on viewing of the moving image content, based on the acquired information; andgenerate the new moving image thumbnail based on a result of the evaluation.5. The server according to claim 4, wherein the processor is further configured to:perform machine learning about a correlation between the index value and information on a user who operates the terminal device; andgenerate a second new moving image thumbnail for which the index value with respect to a user as a distribution destination is expected to meet a predetermined condition, based on a result of the machine learning.6. The server according to claim 4, wherein the processor is further configured to distribute the new moving image thumbnail to the terminal device in accordance with a distribution rate that is calculated based on the result of the evaluation.7. The server according to claim 1, whereinthe communicator acquires, as the information on the usage, at least one of a rate at which a user selects the first thumbnail, a completion rate of a moving image content viewed via the first thumbnail, and a bookmark rate of the moving image content, andthe processor generates the new moving image thumbnail based on the acquired information on the usage.8. A method of composing moving thumbnails tailored to users, implemented by a computer, comprising:acquiring information on usage of a moving image content corresponding to a first thumbnail used as a key image of the moving image content in a terminal device that is a distribution destination of the first thumbnail;acquiring a time point during the first thumbnail when the moving image content is requested;selecting a first portion of the moving image content corresponding to the time point;selecting a second portion of the moving image content based on the acquired information, the second portion of the moving image content being absent from the first thumbnail; andgenerating, using the computer, a new moving image thumbnail in which the second portion of the moving image content is played, and the first portion of the moving image content is played at a middle point or immediately after a start point of the new moving image thumbnail.9. A non-transitory computer readable storage medium having stored therein a generation program causing a computer to execute a process comprising:acquiring information on usage of a moving image content corresponding to a first thumbnail used as a key image of the moving image content in a terminal device that is a distribution destination of the first thumbnail;acquiring a time point during the first thumbnail when the moving image content is requested;selecting a first portion of the moving image content corresponding to the time point;selecting a second portion of the moving image content based on the acquired information, the second portion of the moving image content being absent from the first thumbnail; andgenerating a new moving image thumbnail in which the second portion of the moving image content is played, and the first portion of the moving image content is played at a middle point or immediately after a start point of the new moving image thumbnail.