The main reason for Australia to be interested in the issues faced by the Solomon Islands is geographic. Because the Solomons are so close to Australia, and because Australia is the major regional power, Australia is very interested in the Solomons. Let us look at three major reasons for this.
First, Australia has been connected to the Solomons since World War II. During that war, Australia (along with the US) fought to prevent the Japanese takeover of the Solomons. After that, Australia maintained strong business ties with the Solomons.
Second, Australia is the major economic power in the region. Any country that is a major economic power will naturally want to have some degree of control over its neighbors. It will want to make use of whatever economic resources the neighbors have. For this reason, Australia will naturally want to be involved in the Solomons.
Finally, since Australia is the major economic power in the region, it would absorb the brunt of any major problems that happened in the Solomons. If the Solomons had major problems, people from there would try to emigrate to Australia just as there are many refugees from Asia trying to get to Australia. This would place a great strain on Australia both economically and socially.
For these reasons, it is natural that Australia will take an interest in the problems of the Solomon Islands.