mysql - query records to fetch by time stamp
Since the seconds are from 0 (1970-01-01 00:00:00 UTC), you should look for every multiple of 60

SELECT * FROM mytable WHERE MOD(TimeStamp,60)=0;


or if TimeStamp is indexed, you can do

SELECT T.* FROM
(SELECT TimeStamp FROM mytable WHERE MOD(TimeStamp,60)=0) M
INNER JOIN mytable T USING (TimeStamp);


Give it a Try !!!

SUGGESTION #1

You should store the timestamp of the minute and index it

ALTER TABLE mytable ADD COLUMN MinuteTimeStamp AFTER TimeStamp;
UPDATE mytable SET MinuteTimeStamp = TimeStamp - MOD(TimeStamp,60);
ALTER TABLE mytable ADD INDEX MinuteTimeStamp_UniqueKey_ndx (MinuteTimeStamp,UniqueKey);


Then, you can do MIN aggregation on MinuteTimeStamp.

SELECT MinuteTimeStamp,MIN(UniqueKey) UniqueKey
FROM mytable GROUP BY MinuteTimeStamp;


and use it get those records

SELECT B.* FROM
(SELECT MinuteTimeStamp,MIN(UniqueKey) UniqueKey
FROM mytable GROUP BY MinuteTimeStamp) A
INNER JOIN mytable B USING (UniqueKey);


It was tactfully pointed out that triggers would degrade performance

Perhaps doing INSERTs like this may help

INSERT INTO mytable (UniqueKey,TimeStamp,MinuteTimeStamp) VALUES
(
    uniquevalue,
    UNIX_TIMESTAMP(NOW()),
    UNIX_TIMESTAMP(NOW() - INTERVAL SECOND(NOW()) SECOND)
);


SUGGESTION #2

Since you have over 1000 columns (Ugh), perhaps a table of those minute timestamps would be better.

CREATE TABLE MinuteKeys
(
    MinuteTimeStamp INT UNSIGNED NOT NULL,
    UniqueKey INT UNSIGNED NOT NULL,
    PRIMARY KEY (UniqueKey)
    KEY MinuteTimeStamp_UniqueKey_ndx (MinuteTimeStamp,UniqueKey)
) ENGINE=MyISAM;
ALTER TABLE MinuteKeys DISABLE KEYS;
INSERT INTO MinuteKeys SELECT TimeStamp - MOD(TimeStamp,60),UniqueKey FROM mytable;
ALTER TABLE MinuteKeys ENABLE KEYS;


Then, you could use that table for the aggregation

SELECT B.* FROM
(SELECT MinuteTimeStamp,MIN(UniqueKey) UniqueKey
FROM MinuteKeys GROUP BY MinuteTimeStamp) A
INNER JOIN mytable B USING (UniqueKey);


EPILOGUE

Other suggestions are possible but you should really consider normalization of the table

See my post Too many columns in MySQL as to why