Hi Mark,
First of all – does your cellular plan include/allow international texting? If you’re not sure ask your wireless provider.
The digits you are using look right although I am not aware of Solomon Islands cell numbers starting with 76 (most start with 74 and 75). It is possible of course that they’ve added some new prefixes. Have you been able to call this number? Instead of using the plus sign before 677 you can also use 00 for both texting and calling: 0067776000??.
Let me know if this works.