At first, I think there is not the existing block which meets your need, for your need is someting special.

The only way is to hack the Mylinks module.
1)To get random link, you could use the MySQL sql.
for example, to get 5 random link from table 'linktable' in database, the sql could be:
select * from linktable order by rand() limit 0,5;

2)To get image with text, just modify block template. And it's very easy to be done if you konw about HTML.

Good luck!

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