Way to try multiple SELECTs till a result is available?

LIKE without wildcard character is equivalent to =. Assuming you actually meant name="text".

Indexes are the key to performance.

Test setup

CREATE TABLE image (
  image_id serial PRIMARY KEY
, group_id int NOT NULL
, name     text NOT NULL
);

Ideally, you create two indexes (in addition to the primary key):

CREATE INDEX image_name_grp_idx ON image (name, group_id);
CREATE INDEX image_grp_idx ON image (group_id);

The second may not be necessary, depending on data distribution and other details. Explanation here:

Query

This should be the fastest possible query for your case:

SELECT * FROM image WHERE name="name105" AND group_id = 10
UNION ALL
SELECT * FROM image WHERE name="name105"
UNION ALL
SELECT * FROM image WHERE group_id = 10
LIMIT  1;

SQL Fiddle.

The LIMIT clause applies to the whole query. Postgres is smart enough not to execute later legs of the UNION ALL as soon as it has found enough rows to satisfy the LIMIT. Consequently, for a match in the first SELECT of the query, the output of EXPLAIN ANALYZE looks like this (scroll to the right!):

Limit  (cost=0.00..0.86 rows=1 width=40) (actual time=0.045..0.046 rows=1 loops=1)
  Buffers: local hit=4
  ->  Result  (cost=0.00..866.59 rows=1002 width=40) (actual time=0.042..0.042 rows=1 loops=1)
        Buffers: local hit=4
        ->  Append  (cost=0.00..866.59 rows=1002 width=40) (actual time=0.039..0.039 rows=1 loops=1)
              Buffers: local hit=4
              ->  Index Scan using image_name_grp_idx on image  (cost=0.00..3.76 rows=2 width=40) (actual time=0.035..0.035 rows=1 loops=1)
                    Index Cond: ((name="name105"::text) AND (group_id = 10))
                    Buffers: local hit=4
              ->  Index Scan using image_name_grp_idx on image  (cost=0.00..406.36 rows=500 width=40) (never executed)
                    Index Cond: (name="name105"::text)
              ->  Index Scan using image_grp_idx on image  (cost=0.00..406.36 rows=500 width=40) (never executed)
                    Index Cond: (group_id = 10)
Total runtime: 0.087 ms

Bold emphasis mine.

Do not add an ORDER BY clause, this would void the effect. Then Postgres would have to consider all rows before returning the top row.

Final questions

Is there a generic solution for that?

This is the generic solution. Add as many SELECT statements as you want.

Of course it would come in handy when the search result is sorted by its relevance.

There is only one row in the result with LIMIT 1. Kind of voids sorting.

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