From BiblePortal Wikipedia

King James Dictionary [1]

BESI'DE, prep. be and side, by the side.

1. At the side of a person or thing near as, sit down beside me, or beside the stream. 2. Over and above distinct from.

Beside all this, between us and you, there is a great gulf fixed.  Luke 16 .

3. On one side out of the regular course or order not according to, but not contrary.

It is beside my present business to enlarge upon this speculation.

4. Out of in a state deviating from as, to put one beside his patience. Hence, 5. With the reciprocal pronoun, beside one's self is out of the wits or senses out of the order of reason, or of rational beings.

Paul, thou are beside thyself.  Acts 26 .

Webster's Dictionary [2]

(1): (n.) Over and above; distinct from; in addition to.

(2): (n.) Aside from; out of the regular course or order of; in a state of deviation from; out of.

(3): (adv.) On one side.

(4): (adv.) More than that; over and above; not included in the number, or in what has been mentioned; moreover; in addition.

(5): (n.) At the side of; on one side of.

International Standard Bible Encyclopedia [3]

bē̇ - sı̄d ´: Near to, or close to ( Psalm 23:2 ). It is often used to refer to the mental state, to the derangement of the mind (ἐξίστημι , exı́stēmi ,  Mark 3:21;  Acts 26:24 the King James Version). Or it may refer to the condition of being out of the ordinary course of the life. A life consecrated to God and spent in the interest of humanity is so designated (  2 Corinthians 5:13 ). It has the sense also of a state of being out of one's usual mind, but not of mental derangement, occasioned by something that causes amazement or ast onishment ( Mark 5:42 ). Or it may refer to a state in which one is not conscious of present conditions, but is rapt in vision ( Acts 10:10 ).

Besides is used in the sense of in addition to or that which is over and above what has been said or is possessed (  Luke 16:26; see the American Revised Version, margin "in";  Philippians 1:19 ).